# Calculus: Functions

Consider the function f(x) = x^3 * e^-4x for x is greater than or equal to 0.

(a) Find the maximum value of f(x) on the interval [0,infinity).

(b) Find the minimum if f(x) on the interval [0,infinity).

(c) Find, if there are any, the points of inflection on the interval [0,infinity).

(d) For what values of x on the interval [0,infinity) id f(x) increasing?

(e) For what values of x on the interval [0,infinity) id f(x) concave up?

https://brainmass.com/math/calculus-and-analysis/finding-maximum-minimums-functions-322631

#### Solution Preview

(a) f'(x) = x^3 * -4 e^(-4x) + [e^(-4x)] * 3x^2 = [e^(-4x)] (x^2) (3 - 4x) = 0

x = {0, 3/4}

f"(x) = [e^(-4x)] (6x - 12x^2) + (3x^2 - 4x^3)(-4) e^(-4x) = [e^(-4x)] (2x) (3 - 12x + ...

#### Solution Summary

The expert finds the maximum and minimums of functions. Intervals of increasing and concave up are determined. A complete, neat and step-by-step Solution is provided in the attached file.