Use the Second Derivative Test to find the relative extrema, if any, of the function .

Question

Use the Second Derivative Test to find the relative extrema, if any, of the function  .

Quizplus · Accepted Answer

The second derivative of the function is 22eaa3e5_55d5_8276_9f8f_8a859b3607ec_TB5971_11 which is always positive. Hence, the function is always concave up and any critical point will be a relative minimum. 
Now, to find the relative maximum, we need to look at the behavior of the function around the critical point. The critical point is at x=3. 
Computing the first derivative yields 11eaa3e5_55d5_a27f_9f8f_ed277c5e5db5_TB5971_11 which changes sign from negative to positive as we cross x=3, indicating a local minimum at that point. Therefore, the function has a relative maximum to the left of x=3 and a relative minimum at x=3. 
Using test points to the left and right of x=3 in the first derivative, we can confirm that the relative maximum is at x=1 and has a value of 11eaa3e5_55d4_f74d_9f8f_9be7a9e90cfa_TB5971_11, while the relative minimum at x=3 has a value of 11eaa3e5_55d4_f74e_9f8f_c9175818d242_TB5971_11. Therefore, the answer is B.

Use the Second Derivative Test to Find the Relative Extrema