Question

second derivative test: problem 2
(1 point)
the function f has a continuous second derivative, and it satisfies ( f(-6)=8 ), ( f(-6)=-3 ) and ( f(-6)=-1 ).
we can conclude that
o a. f has a local minimum at -6
o b. f has a local maximum at -6
o c. f has neither a local maximum nor a local minimum at -6
o d. we cannot determine if a, b, or c hold without more information.

Explanation:

Step1: Recall the second - derivative test

The second - derivative test states that if \(f^{\prime}(c)=0\) and \(f^{\prime\prime}(c)>0\), then \(f(x)\) has a local minimum at \(x = c\); if \(f^{\prime}(c)=0\) and \(f^{\prime\prime}(c)<0\), then \(f(x)\) has a local maximum at \(x = c\). Here, \(c=-6\), \(f^{\prime}(-6)=- 3
eq0\).

Answer:

C. \(f\) has neither a local maximum nor a local minimum at \(-6\)