Answer to Question #129484 in Calculus for Jade

Question

Suppose h’’ is continuous at x=2, h’(2)=0, and h’’(2)<0

What can you say about the behavior of h at x=2

A: inflection point B:inflection point at a horizontal tangent C: relative(local) minimum D: relative(local) maximum

Answer 1

The second derivative test of extrema states:

Let $f(x)$ be a function with $f\prime (x_0)=0$. Then if $f\prime\prime(x_0)>0$, the function has a local minimum at $x=x_0$. If $f\prime\prime(x_0)<0$, the function has a local maximum at $x=x_0$ . If $f\prime\prime(x_0)=0$, the second derivative test fails.

The function $h(x)$ satisfies the conditions of the second derivative test at point $x_0=2$ and $h\prime\prime(2)<0$, hence the function $h$ has a local maximum at $x=2$.

Answer. D: relative(local) maximum