Question

Consider the function f(x) = 3x^4 - 8x^3.

a) Determine the inflection points
b) Find the local maxima
c) Identify the critical points
d) Calculate the average rate of change

Answer 1

To analyze the function f(x) = 3x^4 - 8x^3, we must find the second derivative for inflection points, set the first derivative to zero for local maxima, look for points where the first derivative is zero or undefined for critical points, and calculate the average rate of change over an interval.

Step-by-step explanation:

To address the student's question about the function f(x) = 3x^4 - 8x^3, we need to perform several calculations involving derivatives and rates of change. We'll find the inflection points, determine local maxima, identify critical points, and calculate the average rate of change.

1. To find the inflection points, we need to compute the second derivative of f(x) and solve for x when this derivative is equal to zero.
2. Local maxima can be found by setting the first derivative of f(x) to zero and determining if these points are in fact local maxima by analyzing the sign changes in the first derivative or using the second derivative test.
3. Critical points occur where the first derivative of f(x) is zero or undefined.
4. The average rate of change between two points a and b on the function is calculated by the formula (f(b) - f(a)) / (b - a).