For any twice differentiable function f: R - R, if at some x* in R, f'(x*) = 0 and f''(x*) 0, then the function f nec...

Question

For any twice differentiable function f: R -> R, if at some x* in R, f'(x*) = 0 and f''(x*) > 0, then the function f necessarily has a _____ at x = x*.

Note: R denotes the set of real numbers.

A. local minimum
B. global minimum
C. local maximum
D. global maximum

Accepted Answer

The function f necessarily has a LOCAL MINIMUM at x = x*. Answer: A. The second derivative test states that if f'(c) = 0 and f''(c) > 0, then f has a local minimum at x = c. The positive second derivative means the function is concave up at that point, like the bottom of a bowl. However, this does not guarantee a global minimum, because f could have lower values elsewhere.

For any twice differentiable function f: R -> R, if at some x* in R, f'(x*) = 0 and f''(x*) > 0, then the function f necessarily has a _____ at x = x*. Note: R denotes the set of real numbers. A. local minimum B. global minimum C. local maximum D. global maximum

For any twice differentiable function f: R -> R, if at some x* in R, f'(x) = 0 and f''(x) > 0, then the function f necessarily has a _____ at x = x*. Note: R denotes the set of real numbers. A. local minimum B. global minimum C. local maximum D. global maximum