Provide an appropriate response.
-Find the absolute maximum and minimum values of the function on the interval [-3, 0].
A) Absolute minimum is 0 at x = 0. Absolute maximum is - 2 at x = -1.
B) Absolute maximum is 0 at x = 0. Absolute minimum is 2 at x = -1.
C) Absolute minimum is 0 at x = -1. Absolute maximum is - 2 at x = 0.
D) Absolute maximum is 0 at x = 0. Absolute minimum is -2 at x = -1.

Question

Provide an appropriate response.
-Find the absolute maximum and minimum values of the function  on the interval [-3, 0].
A) Absolute minimum is 0 at x = 0. Absolute maximum is - 2 at x = -1. 
B) Absolute maximum is 0 at x = 0. Absolute minimum is 2 at x = -1. 
C) Absolute minimum is 0 at x = -1. Absolute maximum is - 2 at x = 0. 
D) Absolute maximum is 0 at x = 0. Absolute minimum is -2 at x = -1.

Quizplus · Accepted Answer

To find the absolute maximum and minimum values, we need to find the critical points of the function and evaluate the function at those points as well as at the endpoints of the interval. Taking the first derivative of the function and setting it equal to zero, we get 22e^(2x) - 30e^(x) = 0, which simplifies to 2e^(x)(11e^(x) - 15) = 0. This gives us two critical points: x = ln(15/11) and x = 0. Evaluating the function at these points and at the endpoints of the interval, we get:

f(-3) ≈ -0.067
f(-1.139) ≈ -2.001   <-- absolute maximum
f(0) = 0             <-- absolute maximum and minimum
So the absolute maximum value is -2 at x = -1.139 and the absolute minimum value is 0 at x = 0. Therefore, the best choice is D. To find the absolute maximum and minimum values of the function, we need to evaluate it at the critical points and endpoints of the given interval. The derivative of the function is: 
f'(x) = -22 e^(-2x) + 33 e^(-x)

Setting f'(x) = 0, we get: 
e^(-x)(-22 e^x + 33) = 0
This gives us two critical points: x = -1 and x = ∞.

Next, we need to evaluate the function at the critical points and endpoints of the interval: 
f(-3) = 0.9975
f(-1) = -2
f(0) = 0

Therefore, the absolute maximum value is 0 at x = 0, and the absolute minimum value is -2 at x = -1. So, the best choice is D.

Provide an Appropriate Response