4.1 Extreme Values

f(c) can occur where f' (c) =? it can also occur at? (4) this is where f' (c) does not? The function has real solutions when the discriminant is? (3) It can have how many local extrema values? (3) To find extrema from an absolute value equation like y = |x +1| + | 5x-6| : find places where the? these include the? (2)

0, corners/cusps/vertical tangents/discontinuities, exist, positive/negative/0, 0/1/or 2, derivative does not exist, corners/endpoints

The local max can be? the local min can be? At x = 0 for a critical point, it is only an extrema if x exists on the? Given square root of x^2-1 and (1,0) (-1,0), say the absolute extrema is? it exists at? Given [2,7] and the extrema is at x = 5, the interval for the extrema is? When finding the derivative of x • square root of 4-x^2, you get the fraction -2x^2 +4/ square root of 4-x^2. Solve for the? those points are?

<> absolute min. < absolute max, graph f(x), 0, x = 1 and -1, [5, 7], numerator and denominator, critical points

Extreme value theorem? If the interval is open like (a,b), it may not have? Local maximum value: when does it happen? Local minimum: when does it happen? A function has a local max/min at the endpoint c if the inequality is?-ex. (3, 5]. relative extrema def?

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b], absolute extrema, when f(x) <_ f(c) in an open interval, when f(x) >_ f(c) in an open interval, half open, local extrema

Extreme Value Theorem: if f is not continous on [a, b], there might not be? there can still be? Ex. an open circle on f(c) means f is not? Local extrema appear in? ex. (-2, 0]. On a graph, this is an interval bounded by? (2) The absolute min is the? The absolute max is the? Absolute extrema can also be? this is only if the absolute extrema aren't on?

absolute extrema, absolute extrema, continuous on the interval, half-open intervals, 1 open circle/1 closed circle, smallest local extrema, greatest local extrema, local extrema, endpoints

global extrema are? relative extrema are? glpbal max is? Given 2x • e^x - 2e^x /x: factor out a? use? ex. factor our? this method will give you all the? Even if x = 0 is discontinuous on f(x) and is not an extrema, you can still use it to find the? plug in points and test them in the? ex. for x = 0, test the left value? and right value? For y'' with a huge equation, factor out a? use?

absolute extrema, local extrema, absolute max, common term, zero product property, e^x, critical points, intervals of incrrase and decrease, first derivative, -0.5, 0.5, common term, zero product property

Piecewise inequality function: Since it has open intervals, it might not have? it likely has? For an x equation value x >1 and x<_1, check the? Remember to calculate left and right derivative: Plug h +x (ex. h +1) into every? subtract by the? divide by? If the one-sided derivatives are different, no derivative exists and x = 1 is a?

absolute extrema, local extrema, left and right derivative, x variable, f(x) or y value, h, critical point,

For piecewise inequalities: To check if the point is a max or min, sketch the graph and? If an even function has a local max at x = a, x = -a is also a? (0,0): if a point doesn't appear to be a max or min, it is not an? ex. (0,0) is between a higher and lower value. For test: open circles indicate? (2) Given equations with intervals like -7<_ x <_ 5, even if the graph does not appear to have an absolute min, this is a graph of the? not the? Rely on the?

check the points, local max, extrema, parentheses/open interval, interval, whole graph, f(x) values

For piecewise functions and finding extrema values; Find the derivative of each? set =? solve for? Remove x values that don't exist in the? the point on the x interval is a? ex. (2,5) on x <_ 2. The intervals will tell you if there is a? (2) ex. x >2 and x<_ 2, there is only a? no? Discontinuities on the piecewise function are)

equation, 0, x, interval, critical point, max/min, local min, local max, critical points

Discontinuities on the graph (ex. x = 0 is a critical point but discontinuous) are not? To distinguish local extrema from absolute extrema: first, find the? then use the? create # line and test? the highest and lowest extrema are the? Absolute extrema: it can be a? the point doesn't have to be? remember: Extrema values must exist on the?

extrema,critical points, first derivative test, values left and right, absolute extrema, continous point disconnected from the graph, connected to the graph, f(x) graph

Local Extreme Values (Definition): If f has a local max or min at an interior point c, and if f' exists at c, then? Steps to find extrema: First, plug the endpoints a and b into the? Then, find the? set it equal to? solve for? plug x values into? Critical point def.? stationary point def.? Local and absolute extrema can only occur at? (2)

f' (c) = 0, original equation, derivative, 0, x, original equation, point where f' = 0 or does not exist, critical point where f' = 0, endpoints/critical points

For y = 5x • (32-x^2)^1/2 : Instead of zero product property, you can? this is more? The endpoint extrema are? not automatically? Given max endpoint (-2,0) and local max (3, 5), the absolute max is? why? Local extrema for endpoints: for the # line, if the right endpoint is + and increasing on the f' (x) graph, it is a? if it f' is - and decreasing, it is a? Left endpoint: If f' is - and decreasing, it is a? if f' is + and increasing, it is a?

rationalize the denominator and numerator, accurate, local extrema, absolute extrema, (3,5), it has the greatest y value, local max,' local min, local max, local min,

Absolute maximum value: when does this happen? Absolute minimum value: when does this happen? Absolute or global extrema def.? A domain of (-infinity, infinity): the function has no? (0, 2): no absolute? (0,2): no?

when all f(x) values <_ f(c) values, when allf (x) values >_ f(c) values, absolute max and min values, absolute maximum, minimum, absolite extrema

for square root of x^2-36, two absolute minimum values are possible if the? (is the? for the?) ex. (6,0) and (-6,0) can both be? For 1/ (1-9x^2)^ 1/5 : the mini parabola on the graph had a? It is not an absolute value as there are x values...? Remember to identify the domain of a? ex. square root of 64-x^2: domain is [-8, 8]. The domain will give you the?

y value is the same for the x values, absolute mins, local min, outsife the interval, function, endpoints

Given y = square root of 4-x^2: factor out with? this will give you all the? the critical points are usually? (2) Given x^3: to the right, f' (x) is? to the left? If x shows a concavity change but is discontinuous on the graph; it can't be an? Ex. x value on an inequality often has? this creates a? A graph with zero critical points has no? the curve never? A graph with 2 critical points has? a function has 0 real solutions when the discriminant is? it had 1 real solution when the discriminant is? it has 2 real solutions when the discriminant is?

zero product property, critical points, +or - 2 / + or - square root of 2, increasing? decreasing, inflection point, different one-sided limits, discontinuity, extrema, changes, 2 extrema, negative, 0, positive