hw 18 - second derivative test section 3.4: problem 2\n(1 point)\nthe function ( f(x) ) has a continuous…

hw 18 - second derivative test section 3.4: problem 2\n(1 point)\nthe function ( f(x) ) has a continuous second derivative, and it satisfies ( f(-6)=8, f(-6)=0 ) and ( f(-6)=1 ).\nwe can conclude that\n○ a. ( f(x) ) has neither a relative maximum nor a relative minimum at -6\n○ b. ( f(x) ) has a relative minimum at -6\n○ c. ( f(x) ) has a relative maximum at -6\n○ d. we cannot determine if a, b, or c hold without more information.

hw 18 - second derivative test section 3.4: problem 2\n(1 point)\nthe function ( f(x) ) has a continuous second derivative, and it satisfies ( f(-6)=8, f(-6)=0 ) and ( f(-6)=1 ).\nwe can conclude that\n○ a. ( f(x) ) has neither a relative maximum nor a relative minimum at -6\n○ b. ( f(x) ) has a relative minimum at -6\n○ c. ( f(x) ) has a relative maximum at -6\n○ d. 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 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)