Question

hw 18 - second derivative test section 3.4: problem 1
(1 point)
the function ( f(x) ) has a continuous second derivative, and it satisfies ( f(-5)=3, f(-5)=0 ) and ( f(-5)=-2 ).
we can conclude that
a. ( f(x) ) has a relative maximum at -5.
b. ( f(x) ) has neither a relative maximum nor a relative minimum at -5.
c. ( f(x) ) has a relative minimum at -5.
d. we cannot determine if a, b, or c hold without more information.

Explanation:

Step1: Recall the second - derivative test

If \(f^{\prime}(c)=0\) and \(f^{\prime\prime}(c)>0\), then \(f(x)\) has a relative minimum at \(x = c\). If \(f^{\prime}(c)=0\) and \(f^{\prime\prime}(c)<0\), then \(f(x)\) has a relative maximum at \(x = c\). If \(f^{\prime}(c)=0\) and \(f^{\prime\prime}(c)=0\), the second - derivative test is inconclusive.
Here \(c=-5\), \(f^{\prime}(-5) = 0\) and \(f^{\prime\prime}(-5)=-2<0\).

Answer:

A. \(f(x)\) has a relative maximum at \(-5\)