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

To use the Second Derivative Test, we need to find the first and second derivative of the function: 
f(x) = 11e^(ax)

f'(x) = 11ae^(ax) 
f''(x) = 11a^2e^(ax)

To find the critical points, we set f'(x) = 0: 
11ae^(ax) = 0 
a = 0

Therefore, the function has a critical point at x = 0.

To determine the nature of this critical point, we evaluate f''(0): 
f''(0) = 11a^2e^(a(0)) = 11a^2

Since a^2 is always positive, the sign of f''(0) depends on the value of a.

If a is positive, then f''(0) > 0, which means the critical point is a relative minimum. 
If a is negative, then f''(0) < 0, which means the critical point is a relative maximum.

Based on these observations, we can see that the only choice that fits the Second Derivative Test is B: 
Rel.max. 11eaae29_f523_aeeb_87fe_c5eb4bd1a15e_TB2067_11 ,
Rel)min. 11eaae29_f523_d5fc_87fe_2fb4cfbb234d_TB2067_11

In choice B, we have a negative value for a in the first function (which corresponds to a relative maximum) and a positive value for a in the second function (which corresponds to a relative minimum).

Use the Second Derivative Test to Find the Relative Extrema,if