Question

the function $f(x)$ has a continuous second derivative, and it satisfies $f(4)= - 7,f(4)=0$ and $f(4)=1$. we can conclude that
a. $f(x)$ has a local minimum at $x = 4$.
b. $f(x)$ has neither a local maximum nor a local minimum at $x = 4$.
c. $f(x)$ has a local maximum at $x = 4$.
d. we cannot determine if a, b, or c hold without more information.

Explanation:

Step1: Recall second - derivative test

The second - derivative test states that if a function $y = f(x)$ has a continuous second - derivative, and at a point $x = c$, $f'(c)=0$ and $f''(c)>0$, then the function has a local minimum at $x = c$; if $f'(c) = 0$ and $f''(c)<0$, then the function has a local maximum at $x = c$; if $f'(c)=0$ and $f''(c) = 0$, the test is inconclusive.

Step2: Apply the test to the given function

We are given that $f(4)=-7$, $f'(4) = 0$ and $f''(4)=1$. Since $f'(4) = 0$ (critical point) and $f''(4)=1>0$, by the second - derivative test, the function $f(x)$ has a local minimum at $x = 4$.

Answer:

A. $f(x)$ has a local minimum at $x = 4$.