Question

the function ( f ) has a continuous second derivative, and it satisfies ( f(1)=-9 ), ( f^{prime}(1)=0 ) and ( f^{prime prime}(1)=-1 ).
we can conclude that
a. ( f ) has a local maximum at 1.
b. ( f ) has a local minimum at 1.
c. ( f ) has neither a local maximum nor a local minimum at 1.
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 \(f^{\prime\prime}(c)\) exists:

• If \(f^{\prime\prime}(c)>0\), then \(f(x)\) has a local minimum at \(x = c\).
• If \(f^{\prime\prime}(c)<0\), then \(f(x)\) has a local maximum at \(x = c\).
• If \(f^{\prime\prime}(c)=0\), the test is inconclusive.

Step2: Apply the second - derivative test

Given \(c = 1\), \(f^{\prime}(1)=0\) (critical point) and \(f^{\prime\prime}(1)=- 1<0\).

Answer:

A. \(f\) has a local maximum at \(1\).