Question

the function f(x) has a continuous second derivative, and it satisfies f(3)=7, f(3)=12 and f(3)=0. we can conclude that
o a. f(x) has a local maximum at x = 3.
o b. f(x) has a local minimum at x = 3.
o c. f(x) has neither a local maximum nor a local minimum at x = 3.
o d. we cannot determine if a, b, or c are true without more information.

Explanation:

Step1: Recall the first - derivative test conditions

For a function $y = f(x)$ to have a local extremum at $x = c$, $f'(c)=0$. Here, $f'(3)=12
eq0$.

Step2: Recall the second - derivative test conditions

The second - derivative test is used when $f'(c) = 0$. If $f'(c)=0$ and $f''(c)>0$, then $f(x)$ has a local minimum at $x = c$; if $f'(c)=0$ and $f''(c)<0$, then $f(x)$ has a local maximum at $x = c$. But since $f'(3)
eq0$, the second - derivative test is not applicable.

Answer:

C. $f(x)$ has neither a local maximum nor a local minimum at $x = 3$.