05.05 Determining Concavity of Functions and the Second Derivative Test

The second derivative of function g is given by g double prime of x equals 2 times x all over the quantity 1 plus x squared all squared end quantity period For which interval is the graph of g concave up?

(0, infinity)

The derivative of function f is given by f prime of x equals negative 2 times x where x is less than 1 and equals x squared plus 2 where x is greater than or equal to 1 period For which interval is the graph of f concave up?

(1, infinity)

Selected values of f ′ and f ″ are shown in the table for the twice-differentiable function f: x 0 0.5 1 1.5 2 2.5 3 f ′(x) undefined −8 −3−1.3333 −0.5 0 0.3333 f″(x) undefined 20 5 2.2222 1.25 0.8 0.5556 Which of the following is true, based on the information in the table?

f has a relative minimum at x = 2.5

The function f(x) is continuous over the interval [−2, 5] and has a critical point at x = 1. If f ″ is negative on this interval, which of the following is true?

f has an absolute maximum at x = 1

The second derivative of the function h is given by h″(x) = 5x3 − 15x. At what value(s) of x does the graph of h have a point of inflection? (1 point)

x equals negative square root of 3, x equals 0, and x equals square root of 3