the function f has a continuous second derivative, and it satisfies f(-9)=8, f(-9)=0 and f(-9)=0. we can…

the function f has a continuous second derivative, and it satisfies f(-9)=8, f(-9)=0 and f(-9)=0. we can conclude that\na. f has a local maximum at -9\nb. f has neither a local maximum nor a local minimum at -9\nc. f has a local minimum at -9\nd. we cannot determine if a, b, or c hold without more information.

the function f has a continuous second derivative, and it satisfies f(-9)=8, f(-9)=0 and f(-9)=0. we can conclude that\na. f has a local maximum at -9\nb. f has neither a local maximum nor a local minimum at -9\nc. f has a local minimum at -9\nd. we cannot determine if a, b, or c hold without more information.

Answer

Explanation:

Step1: Recall the second - derivative test

The second - derivative test for local extrema states that if (f^{\prime}(c)=0) and (f^{\prime\prime}(c)>0), then (f) has a local minimum at (c); if (f^{\prime}(c) = 0) and (f^{\prime\prime}(c)<0), then (f) has a local maximum at (c). But if (f^{\prime}(c)=0) and (f^{\prime\prime}(c) = 0), the second - derivative test is inconclusive.

Step2: Analyze the given values

We are given that (f(-9)=8), (f^{\prime}(-9)=0) and (f^{\prime\prime}(-9)=0). Since (f^{\prime\prime}(-9) = 0), the second - derivative test (which is based on the sign of the second derivative when the first derivative is zero) cannot be used to determine if (x=-9) is a local maximum, local minimum or neither.

Answer:

D. We cannot determine if A, B, or C hold without more information.