the function ( f ) has a continuous second derivative, and it satisfies ( f(1)=-9 ), ( f^{prime}(1)=0 ) and…

the function ( f ) has a continuous second derivative, and it satisfies ( f(1)=-9 ), ( f^{prime}(1)=0 ) and ( f^{prime prime}(1)=-1 ).\nwe can conclude that\na. ( f ) has a local maximum at 1.\nb. ( f ) has a local minimum at 1.\nc. ( f ) has neither a local maximum nor a local minimum at 1.\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(1)=-9 ), ( f^{prime}(1)=0 ) and ( f^{prime prime}(1)=-1 ).\nwe can conclude that\na. ( f ) has a local maximum at 1.\nb. ( f ) has a local minimum at 1.\nc. ( f ) has neither a local maximum nor a local minimum at 1.\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 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).