Question

hw 18 - second derivative test section 3.4: problem 2
(1 point)
the function ( f(x) ) has a continuous second derivative, and it satisfies ( f(-6)=8, f(-6)=0 ) and ( f(-6)=1 ).
we can conclude that
○ a. ( f(x) ) has neither a relative maximum nor a relative minimum at -6
○ b. ( f(x) ) has a relative minimum at -6
○ c. ( f(x) ) has a relative maximum at -6
○ 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 the function \(f(x)\) has a relative minimum at \(x = c\). If \(f^{\prime}(c)=0\) and \(f^{\prime\prime}(c)<0\), then the function \(f(x)\) has a relative maximum at \(x = c\). If \(f^{\prime}(c)=0\) and \(f^{\prime\prime}(c)=0\), the second - derivative test is inconclusive.

Step2: Apply the second - derivative test

We are given that \(c=-6\), \(f^{\prime}(-6) = 0\) and \(f^{\prime\prime}(-6)=1>0\).

Answer:

B. \(f(x)\) has a relative minimum at \(-6\)