hw 18 - second derivative test section 3.4: problem 1\n(1 point)\nthe function ( f(x) ) has a continuous…

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

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

Answer

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)