precalc unit 1 and unit 2 study guide

The polynomial function p is given by p(x) = (x+2)^4. Which of the following expressions is equivalent to (x+2)^4?

c.) x^4 + 8x^3 + 24x^2 + 32x + 16

The function g is given by g(x) = ln(3x+1) - ln(x^2+x-2). What are all values of x for which g(x) < 0?

d.) (3, ∞) only

The polynomial function p is given by p(x) = (x+3)(x^2 - 2x - 15). Which of the following describes the zeros of p?

a.) p had exactly two distinct real zeros

The polynomial function f is given by f(x) = ax^4 + bx^3 + cx^2 + dx + k, where a ≠ 0 and b, c, d, and k are constants. Which of the following statements about f is true?

b.) f has either a global maximum or a global minimum, but not both

The fifth term of a geometric sequence is 24, and the sixth term is 48. What is the value of the tenth term?

c.) 768

The table gives the average of rates of change of a function f over different intervals. On which of the intervals does the function increase the most?

d.) 8 ≤ x ≤ 10

An equation involves the expression logsub9(27^x), which is equivalent to a rational multiple of x. By rewriting the expression in an equivalent form, the value of the rational number can be determined without use of a calculator or complicated calculations. Which of the following is na equivalent expression that satisfies this requirement?

d.) xlogsub3(27)/logsub3(9)

At a bakery, the number of cookies baked each day changes based on anticipated demand. The scatterplot shows the change in hundreds of cookies baked from the previous day for eight days. The point at (2,5) means that on day 2, the number of cookies baked will be 500 more than the number of cookies baked on day 1. A function model C is to be constructed for the number of cookies baked on each of the days, 0 through 8. Which of the following statements best supports the selection of a model for C?

b.) Because the information about rate of change is roughly linear, a quadratic model is best for C

The rational function g is given by g(x) = (x^3+1000)/(x^2-100) = ((x+10)(x^2-10z+100))/(x^2-100). Which of the following statements describes the behavior of the graph of g?

b.) The graph has hole at x = -10 because (x+10) appears exactly once, when both the numerator and the denominator of g are factored.

The function f is given by f(x) = 3^x. The function g is given by g(x) = (f(x))^b, where b < 0. Which of the following describes the relationship between the graphs of f and g?

b.) The graph of g is a combination of a horizontal dilation of the graph of f and a reflection over the y-axis.

The range of function f is the positive real numbers. The function g is given by g(x) = ln(f(x)). Solutions to which of the following equations are useful in solving g(x) = 2?

b.) f(x) = e^2

Let k, w, and z be positive constants. Which of the following is equivalent to logsub10(kz/w^2)?

b.) logsub10(k) + logsub10(z) - 2logsub10(w)

The daily high temperature at a certain point in a river is modeled by the graph. Each point on a vertical gridline indicates the temperature, in degrees Celsius, on the first day of the indicated month. Of the following, on the first day of which month is the rate of change of the temperature the greatest?

b.) may

The average rates of change of the quadratic function p is -4 on the interval 0 ≤ x ≤ 2 and -1 on the interval 2 ≤ x ≤ 4. What is the average rate of change of p on the interval 6 ≤ x ≤ 8?

c.) 5

The leading term of the polynomial function p is asubnx^n, where asubn is real number and n is positive integer. The factors p include (x-3), (x-i), and (x-(2+i)). What is the least possible value of n?

c.) 5

A ball is thrown through an open window to the ground below. The height of the ball, in meters, at time t seconds after it is thrown can be modeled by the function h, given by h(t) = -4.9t^2 +4.4t + 15.24. Which of the following describes the height of the ball above the ground?

c.) After leaving the window, the height of the ball increases to its maximum height of 16.228 meters. Then the height of the ball decreases until it reaches the ground 1.820 seconds after reaching its maximum height.

The function Q is a polynomial of degree 3. If Q(5) = 0, which of the following must be true?

c.) Q(x) can be expressed as (x-5)(P(x)), where P(x) is a polynomial of degree 2.

A family needs to buy one shovel and between one and eight plants, inclusive, for their new garden. The cost of the shovel is s dollars, and the cost of one plant is p dollars. The output values of which of the following give the possible costs for these items, in dollars?

c.) The arithmetic sequence Csubn = s + pn for 1 ≤ n ≤ 8

A polynomial function has three distinct zeros each with multiplicity 1, and its leading coefficient is positive. The polynomial function has exactly one zero with multiplicity 3, and its leading coefficient is negative. The rational function h can be written as the quotient of p and q by h(x) = P(x)/q(x). Which of the following statements about h must be true?

c.) The graph of h has a horizontal asymptote at y = a, where a < 0.

The function f is given by f(x) = (x+3)^4. When f is rewritten in the form f(x) = x^4 + ax^3 + bx^2 + cx + d, which of the following values is greatest?

c.) c

The function f is given by f(x) = 4(2)^(x-3). If the function g is the inverse of f, which of the following could define g(x)?

c.) logsub2(x/4) + 3

The polynomial function p is an odd function. If p(3) = -4 is a relative minimum of p, which of the following statements about p (-3) must be true?

c.) p(-3) = 4 is a relative minimum.

The graphs of the polynomial functions f and g are shown. The function h is defined by h(x) = f(x)/g(x). What are all vertical asymptotes of the graph of y = h(x)?

c.) x = -2 and x = 3 only

In the xy-plane, the graph of a rational function f has a vertical asymptote at x = -5. Which of the following expressions could define f(x)?

d.) ((x-5)(x-3))/((x-3)(x+5))

The rational function r is given by r(x) = (x^3 +4x^2 +4x)/(x^2-9). On what intervals of x is r(x) ≥ 0?

d.) -3 < x ≤ 0 and x > 3

The table gives values of the function f for selected values of x. If the function f is linear, what is the value of f (13)?

d.) 34/3

An exponential function G has a known common ratio of 1/2 and includes the input-output pair (1,4). Which of the following could define G(x)?

d.) 4(1/2)^(x-1)

The value, in millions of dollars, of transactions processed by an online payment platform is modeled by the function M. The value is expected to increase by 6.1% each quarter of a year. At time t = 0 years, 54 million dollars of transactions were processed. If t is measured in years, which of the following is an expression for M(t)?

d.) 54(1.061)^4t

The functions and are defined for all real numbers such that g(x) = f(2(x-4)). Which of the following sequences of transformations maps the graph of f to the graph of g in the same xy-plane?

d.) A horizontal dilation of the graph of f by a factor of 1/2, following by a horizontal translation of the graph of f by 4 units.

A water tank is leaking water from a crack in its base. The amount of water, in hundreds of gallons, remaining in the tank t hours after the crack formed can be modeled by W, a decreasing function of time t. Which of the following gives a verbal representation of the function W^-1, the inverse function of W?

d.) W^-1 is a decreasing function of the amount of water in the tank

The table gives values of a function f for selected values of x. Which of the following conclusions with reason is consistent with the data in the table?

d.) f could be a quadratic function because the rates of change over consecutive equal-length intervals in the table can be described by y = 2x +1.

The exponential function f is defined by f(x) = ab^x, where a and b are positive constants. The table gives values of f(x) at selected values of x. Which of the following statements is true?

d.) f demonstrates exponential growth because a > 0 and b > 1.

The function f is an increasing function such that every time the output values of the function f increase by 1, the corresponding input values multiply by 4. Which of the following could define f(x)?

d.) logsub4(x)

To solve the equation logsub8(x-3) + logsub8(x+4) = 1, one method is to apply the properties of logarithms to rewrite the equation in an equivalent form. This equivalent equation can be used to identify possible solutions. Of the following, which is such an equation?

d.) x^2 + x - 12 = 8

Let x and y be positive constants. Which of the following is equivalent to 2lnx-3lny?

a.) ln(x^2/y^3)

The exponential function g is given by g(x) = 5^x. Which of the following expressions defines g^-1(x)?

a.) logsub5(x)

The table gives values of the function f for selected values of x. Which of the following is a verbal representation of f^-1(x), the inverse function of f?

b.) f^-1(x) is logarithmic with output values increasing by 1 every time input values double

A decibel (dB) is a unit of measure for loudness of sound. The decibel scale is based in sound intensity N, in watts per square meter. A decibel value is given by the function d, where d(N) = 10logsub10(N/10^-12). Which of the following gives all intensities N, in watts per square meter, for which the decibel value is greater than 140 decibels?

c.) N > 100

The table gives ordered pairs (x, lny). For the function y = f(x), which of the following statements about f is supported by the data in the table?

c.) The function f is exponential because the values of x and and the values of lny both form arithmetic sequences.

The depth of water, in feet, at a certain place in a lake is modeled by a function W. The graph of y = W(t) is shown for 0 ≤ t ≤ 30, where t is the number of days since the first day of a month. What are all intervals of t on which the depth of water is increasing at a decreasing rate?

a.) (3,6) only

The table gives values for the functions and at selected values of x. Functions f and g are defined for all real numbers. Let h be the function defined by h(x) = f(g(x)). What is the value of h(0)?

a.) -2

The first term of an arithmetic sequence is 5, and the common difference of the sequence is 2. What is the eighth term of the sequence?

a.) 19

Two drones are flying over a given area, and their heights above the ground are changing. The table gives the change in height, in feet, for the drones over successive 6-second intervals. Which of the following is true about the average rates of change for drone A and drone B over the time interval from t = 0 seconds to t = 30 seconds?

a.) The average rates of change are equal.

Which of the following names a function with a hole in its graph at x = 1 and provides correct reasoning related to the hole?

a.) The graph of f(x) = (x^2 - 1)/(x-1) has a hole at (1,2) because the values of (x^2 - 1)/(x-1) get arbitrarily close to 1, but the function is undefined at x = 1.

Consecutive terms of a sequence have the values of 6, 2, -2, and -6. Of the following, which describes the sequence?

a.) The terms could be part of an arithmetic sequence with a common difference of -4

The function f has domain [-2,2] and range [1,5]. The function g is given g(x) = -2f(x+3) + 4. What are the domain and range of g?

a.) domain: [-5,-1], range: [-6,2]

The function p (not shown) is a polynomial function of degree 3. The graphs of four functions f, g, h, and k are given. The output values of p are the same as the output values of the composition function when p is composed with one of these functions as the input function. For which of the functions is this statement true?

a.) f

The table gives characteristics of the rates of change of the function f on different intervals. Which of the following is true about f on the interval 3 < x < 4? (positive and decreasing)

a.) f is increasing, and the graph of f is concave down.

The functions f and g are given by f(x) = 4^(5x-1) and g(x) = 8^(x/4). When solving the equation f(x) = g(x), the functions can be rewrittten in equivalent forms so that the equation can be solved without the use of technology. Which of the following are equivalent definitions of f and g that aid in solving f(x) = g(x) without the use of technology?

a.) f(x) = 2^(logsub2(4)(5x-1)) and g(x) = 2^(logsub2(8)*x/4))

The rational function h is expressed as the quotient of two polynomial functions f and g by h(x) = f(x)/g(x). The function f is given by f(x) = 6x^3 + x^2 + 60x - 25. If the graph of has a slant asymptote of y = 2x - 1, which of the following describes g?

a.) g has degree 2 with leading coefficient 3

Which of the following functions has a zero at x = 3 and has a graph in the xy-plane with a vertical asymptote at x = 2 and a hole at x = 1?

a.) h(x) = (x^2-4x+3)/(x^2-3x+2)

The function g is given by g(x) = alogsubb(c), where a, b, and c are positive integers. Which of the following is an equivalent representation of g(x)?

a.) logsubb(c^a)

The function f is given by f(x) = ac^x, where a > 0 and c > 1. Which of the following is true about the values of constants m and b in the equation ln(f(x)) = mx + b?

a.) m > 0 because lnc > 0; b can be any real number because lna can be any real number.

The graph of the function y = g(x) is given. Of the following, on which interval is the average rate of change of g least?

b.) -1 ≤ x ≤ 0

The rational function g is given by g(x) = ((x^2+3x)(x^2-4x-5))/((x+3)(x-1)(x-2)). For what input values of g are the output values of g equal to 0?

b.) -1, 0, and 5 only

The sales of a new product, in items per month, is modeled by the expression 225 + 500logsub10 (15t + 10), where t represents the time since the product became available for purchase in months. What is the number of items sold per month for time = 6?

b.) 1225

The function f is given by f(x) = 2logsub5(x). Which of the following describes f?

b.) f is an increasing function that increases at a decreasing rate.

The graph of the polynomial function g is shown. The function f is defined for 0 ≤ x ≤ 3 and is identical to the function g on that interval. How many total local minima and local maxima does the function f have?

b.) four

In a certain town, the population in the year 2000 as about 30,000. The population grows at a rate of 2.3% per year, and time is measured in years since 2000. Which of the following functions gives output values, in years since 2000, for input values of the town's population p?

b.) g(p) = logsub1.023(p/30,000)

A polynomial function has the form p(x) = ax^j + bx^k, where a and b are nonzero constants, j and k are nonnegative integers. Which of the following conditions guarantees that p is an even function?

b.) j = 2k and k is even

The function f is given by f(x) = 3x^2 + 2x + 1. The graph of which of the following functions is the image of the graph of after a vertical dilation of the graph of f by a factor of 2?

b.) k(x) = 6x^2 + 4x + 2, because this is a multiplicative transformation of f that results from multiplying f(x) by 2

The graph of the polynomial function f is shown. How many points of inflection does the graph of f have one the given portion of the graph?

b.) three

The function f is defined by f(x) = sqrt(4-x^2) for -2 ≤ x ≤ 0. Which of the following expressions defines f^-1(x)?

c.) -sqrt(4-x^2) for 0 ≤ x ≤ 2

The figure shown is the graph of a polynomial function g. Which of the following could be an expression for g(x)?

c.) 0.25(x-5)^2(x-1)(x+8)

For an arithmetic sequence Ssubn, Ssub3 = 3 and Ssub6 = 24. What is the value of Ssub10 - Ssub8

c.) 14

The functions f and g are given by f(x) = x^2 + 1 and g(x) = 4x - 1. Which of the following is an expression for f(g(x))?

c.) 16x^2 - 8x + 2

The table describes rates of change of a function f for selected intervals of x. The function f is defined for 0 ≤ x ≤ 4. On which of the following intervals is the graph of f concave down.

c.) 2 < x < 3

The number of thousands of people that have visited a new website is recorded every 10 days for 60 days. These data are used to produce a semi-log plot as shown. The function N gives the number of thousands of people that have visited the website for day t. Which of the following could define N(t)?

c.) 2.5(2)^(t/10)

The general term of a sequence is given by asubn = 51 + 3 (n-10), where asub0 is the initial value. Which of the following expressions also gives the general term of the sequence?

c.) 21 + 3n

The function f is logarithmic, and the points (2,1) and (4,2) are on the graph of f in the xy-plane. Which of the following could define f(x)?

c.) 2logsub4(x)

The table gives values of a polynomial function Q for selected values of x. What is the degree of Q?

c.) 4

The graph of he piecewise-linear function f is shown in the figure. Let g be the inverse function of f. What is the maximum value of g?

c.) 5

Let f be a rational functional that is graphed in the xy-plane. Consider x = 1 and x = 7. The polynomial in the numerator of f has a zero at x = 1 and does not have a zero at x = 7. The polynomial in the denominator of f has zeros at both x = 1 and x = 7. The multiplicities of the zeros at x = 1 in the numerator and in the denominator are equal. Which of the following statements is true?

c.) The graph of f has a hole at x = 1 and a vertical asymptote x = 7.

The function g is a function of the form g(x) = ab^x where a ≠ 0 and b > 0. The function f is given by f(x) = g(x) + 4. Which of the following statements is true?

c.) The output values of g only, not f, are proportional over equal-length input-value intervals.

The function f is given by f(x) = x^2 +2. the function g is the result of a transformation of f and is given by g(x) = x^2/4 + 2. Which of the following describes the transformation of the graph of f whose image is the graph of g?

c.) a horizontal dilation by a factor of 2

The functions f and g are given by f(x) = 1/x and g(x) = sqrt(x). What is the domain of the function h given by h(x) = f(g(x))

c.) all real numbers greater than 0

The function g is given by g(x) = x^3 - 3x^2 - 18x, and the function h is given by h(x) = x^2 - 2x - 35. Let k be the function given by k(x) = h(x)/g(x). What is the domain of k?

c.) all real numbers x where x ≠ -3, x ≠ 0, x ≠ 6

The function g is given by g(x) = (4x+6)/5. Which of the following defines g^-1(x)?

d.) (5x-6)/4

The function f is not explicitly given. The function g is given by g(x) = f(x+1) - f(x). The function h is given by h(x) = g(x+1) - g(x). If h(x) = -6 for all values of x, which of the following statements must be true?

d.) Because h is negative and constant, g is decreasing, and the graph of f is concave down.

The function f is defined for all real values of x. For a constant a, the average rate of change of f from x = a to x = a +1 is given by the expression 2a + 1. Which of the following statements is true?

d.) The average rate of change of f over consecutive equal-length input-value intervals is increasing at a constant rate, so the graph of f could be a parabola that opens up.

Which of the following includes the input-output pairs (2,4) and (3,8)?

d.) The exponential function h(n) = 2(2)^(n-1)

For the polynomial function g, the rate of change of g is increasing for x < 2 and decreasing for x > 2. Which of the following must be true?

d.) The graph of g has point of inflection at x = 2, is concave up for x < 2, and is concave down for x > 2.

The functions f and g are given by f(x) = 2^x and g(x) = (2^x)(2^a), where a > 0. Which of the following describes the relationship between the graph of f and the graph of g?

d.) The graph of g is a horizontal translation of the graph of f by -a units.

The graph of the function f is given for -3 ≤ x ≤ 6. Which of the following statements about the rate of change of f over the interval 2 < x < 6 is true?

d.) The rate of change is decreasing

The functions g and f are given by g(x) = 3x^2 - 2x and f(x) = 6x^4 + 5x^3 + 3x - 5. Which of the following statements is true about the remainder when f(x) is divided by g(x)?

d.) The remainder is (7x -5), so g(x) is not a factor of f(x), and the graph of y = f(x)/g(x) does not have a slant asymptote.

The ancient Pythagoreans studied figurate numbers, which are numbers that can be shown by taking dots or spheres and arranging them into geometric shapes. For example, the square numbers are 1, 4, 9, 16, 25, etc., and each of these numbers of dots can be arranged into a square. The tetrahedral numbers similarly specify the number of can be arranged into a square. The tetrahedral numbers similarly specify the number of spheres needed to create a tetrahedron, which is a triangular-based pyramid The tetrahedral numbers are 1, 4, 10, 20, 35, 56, 84, etc. Which of the following statements is true?

d.) The tetrahedral numbers are best modeled by a cubic function because the 3rd differences are a nonzero constant.

The function f is given by f(x) = 9(25)^x. Which of the following is an equivalent form for f(x)?

d.) f(x) = 9(5)^2x

Which of the following is the inverse of the function f given by f(x) = 4logsub2(x=3) - 1?

d.) g(x) = 2^((x+1)/4) - 3

The function has a negative average rate of change on every interval of in the interval 0 ≤ x ≤ 10. The function g has a negative average rate of change on every interval of x in the interval 5 < x ≤ 10. Which of the following statements must be true about the function h, defined by h(x) = f(x) + g(x), on the interval 0 ≤ x ≤ 10?

d.) h is decreasing on 0 ≤ x < 4; h can be increasing, decreasing, or both increasing and decreasing on 5 < x ≤ 10.

The function f is given by f(x) = logsub2(x). Which of the following is equivalent to f(7)?

d.) logsub3(7)/logsub3(2)

Iodine-131 has a half-life of 8 days. In a particular sample, the amount of iodine-131 remaining after d days can be modeled by the function h given by h(d) = Asub0(0.5)^(d/8), where Asub0 is the amount of iodine-131 in the sample at time d=0. Which of the following functions k models the amount of iodine-131 remaining after t hours, where Asub0 is the amount of iodine-131 in the sample at time t = 0?

k(t) = Asub0(0.5^(1/192))^t

A polynomial function p is given by p(x) = -x(x-4)(x+2). What are all the intervals on which p(x) ≥ 0?

(-∞, -2] U [0,4]

The function m is given by m(x) = 36^(x/2). Which of the following expressions could also define m(x)?

6^x

In 2016, the cost to mail a package was $2.54 for up to ounces, plus an additional cost of $0.20 for each additional ounce or portion of an ounce less than a full ounce. A portion of the graph of this relationship is given with cost, in dollars, as a function of ounces. Which of the following describes the restrictions on the range for such a function?

D.) The range is values of the form 2.54 + 0.2x, where x is a nonnegative integer.

The function f is given by f(x) = logsub2(logsub3(x)). Which of the following is an expression for f^-1(x)?

b.) 3^2^x

If m = logsub3(81), which of the following is also true?

b.) 3^m = 81

In the xy-plane, the graph of the rational function f has a vertical asymptote at x = 4. Which of the following expressions could define f(x)

a.) ((x+1)^3(x-4)^2)/((x+1)^2 (4-x)^3)

What are all the values of x for which ln(x^3) - lnx = 4?

c.) x = e^2 only

The function f is given by f(x) = logsub10(x). The function g is given by g(x) = logsub10(x^3). Which of the following describes a transformation for which the graph of g is the image of the graph of f?

a.) A vertical dilation by a factor of 3.

The function g is given by g(x) = logsub7(x), and the function h is given by h(x) = logsub49(x). Which of the following describes the relationships between g and h?

a.) For equal input values, the output values of h are half of the output values of g

The function f is given by f(x) = lnx. which of the following describes input values for which the output values of f are integers?

a.) integer powers of e

The logarithmic function f is defined by f(x) = logsub3(x) on a domain of f is 0 < x < ≤ 9. Which of the following is true of f?

b.) f has a maximum value, but no minimum value.

The function f is given by f(x) = 2^x, and the function g is given by g(x) = f(x)/8. For which of the following transformations is the graph of g the image of the graph of f?

b.) A horizontal translation to the right 3 units

In a semi-log plot, which of the following pairs of functions appear linear as parallel lines?

c.) f(x) = 2^x and g(x) = 3(2)^x

The function f is defined by f(x) = 4x^2 + 3 for x ≥ 0. Which of the following expressions defines the inverse function of f?

c.) f^-1(x) = sqrt((x-3)/4) for x ≥ 3

The table gives the average rates of change for the function f, g, h, and k for certain intervals of x. Which of the functions is best modeled by a piecewise-linear segments with different slopes?

c.) h

The initial population size of an animal species is measured to be 2000. The population doubles every 8 years. Which of the following functions gives the time, in years, as an output value, and a certain number x for the population size as an input value?

c.) h(x) = 8logsub2(x/2000)

The function h is given by h(x) = logsub3(x). Which of the following is equivalent to the expression 2(h(w)) + h(p), where w and p are values in the domain of h?

c.) logsub3((w^2)p)

A music agent is planning a series of concerts at local farms around the nation. The agent is building a model to estimate crowd capacity based on different sizes of square fields. Which type of function is most likely to model crowd capacity in this situation?

c.) quadratic

The function h is given by h(x) = 5(3)^(-x/2). What is the value of h(1)?

d.) 5/sqrt(3)

The polynomial function f is given by f(x) = (x-4)(3x-1)^2. Which of the following descriptions of f is true?

d.) f is a polynomial of degree 3 with a leading coefficient of 9.

Water hyacinth is an invasive plant species found in many lakes that typically grows at a rate of 7% per day. As part of a study, a scientist introduces a 150-gram sample of water hyacinth into a testing pool. Which of the following functions gives the amount of water hyacinth in the testing pool t weeks after the sample is introduced?

d.) k(t) = 150(1.07^7)^t

The terms of the increasing arithmetic sequence asubn are positive. The terms of the increasing geometric sequence gsubn are positive. The values of the first terms of both sequences are the same, and the values of the fourth terms of both sequences are the same. Which of the following statements describes the values of the second terms of the sequences?

b.) The second term of the arithmetic sequence must be greater than the second term of the geometric sequence

Consecutive terms of a sequence have the values 2, -1, 1/2, -1/4, and 1/8. Of the following, which describes the sequence?

b.) The terms could be part of a geometric sequence with a common ratio of -1/2.

The figure shows the graph of a function f, the zero and extrema for f are labeled, and the point of inflection of the graph of f is labeled. Let A, B, C, D, and E represent the x-coordinates at those points. Of the following, on which interval is f increasing and the graph of f concave down?

a.) the interval from A to B

The speed of a car traveling on a highway is being recorded once per second for two minutes. During this time interval, the car gradually speeds up slightly to pass another vehicle, then the car returns to its original speed. The recorded speed of the car with respect to time can be modeled by linear, quadratic, and exponential functions. For each of the three models, their residuals are small and are without pattern. Which of the following conclusions is best?

b.) A quadratic model is best based on contextual clues

Values of the polynomial function f for selected values of x are given in the table. If all of the zeros of the function f are given in the table, which of the following must be true?

b.) The function f has a local minimum at (5,0)

The second term of a sequence is 6, and the fourth term is 24. Of the following, which statement is true?

c.) If the sequence is geometric, the fifth term could be 48.