Question

Find the exact values of the absolute maximum and absolute minimum of the function f(x)=x⁴ e⁻³ˣ on the interval [0,2]

Answer 1

To find the exact values of the absolute maximum and absolute minimum of the function f(x)=x⁴ e⁻³ˣ on the interval [0,2], you can use the first and second derivative tests.

Step-by-step explanation:

To find the exact values of the absolute maximum and absolute minimum of the function f(x)=x⁴e⁻³ˣ on the interval [0,2], we can use the first and second derivative tests.

1. Find the critical points by setting the derivative equal to zero: f'(x) = 4x³e⁻³ˣ - 3x⁴e⁻³ˣ = x³e⁻³ˣ(4 - 3x) and solve for x.
2. Evaluate f(0) and f(2) to determine the function values at the endpoints of the interval.
3. Calculate the second derivative f''(x) and evaluate it at the critical points and endpoints.
4. Compare the function values and second derivative values to determine the absolute maximum and minimum.

The absolute maximum and absolute minimum values occur at the critical points and the endpoints.