Use the Second Derivative Test to find the relative extrema, if any, of the function .
A)Rel.max.
B)None
C)Rel.max. , rel.min.
D)Rel.max. , rel.min.

Question

Use the Second Derivative Test to find the relative extrema, if any, of the function  .
A)Rel.max. 
B)None 
C)Rel.max. , rel.min. 
D)Rel.max. , rel.min.

Quizplus · Accepted Answer

First, we need to find the critical points by setting the first derivative to zero:
11e^(3x) - 11e^(x) = 0
11e^(x)(e^(2x) - 1) = 0
e^(2x) = 1 => x = 0 (since e^(2x) is always positive)
So, we have one critical point at x = 0.
Now, we need to find the second derivative:
f''(x) = 33e^(3x) - 11e^(x)
At x = 0: f''(0) = 33 - 11 = 22, which is positive.
Therefore, by the Second Derivative Test, we have a relative minimum at x = 0.
So, the answer is C: Rel.max. 11eaa3e5_55d5_9395_9f8f_f5d9a3b99693_TB5971_11 , rel.min. 11eaa3e5_55d5_9396_9f8f_ddc456abe18c_TB5971_11.

Use the Second Derivative Test to Find the Relative Extrema