Question

Given \( f(x)=-4 x³+8 x²+60 x )

a.) Find the critical points and the intervals on increase and decrease. b.) State whether each critical point is a maximum or a minimum.

Answer 1

To identify the critical points and intervals of increase and decrease of the function f(x) = -4x³ + 8x² + 60x, one must calculate its derivative, set the derivative equal to zero, and use the First Derivative Test to classify the type of each critical point.

Step-by-step explanation:

Finding Critical Points and Analyzing Intervals of Increase and Decrease

To find the critical points of the function f(x) = -4x³ + 8x² + 60x, we first need to calculate its derivative, f'(x). The derivative of the function represents the slope of the tangent line to the curve at any point x, or the rate of change of the function. Therefore, critical points occur where f'(x) = 0 or is undefined. After finding the critical points, we can determine the intervals of increase and decrease by testing values in the intervals around the critical points.

To state whether each critical point is a maximum or a minimum, we can use the First Derivative Test. This involves checking the sign of the derivative before and after each critical point to see if the function changes from increasing to decreasing (indicating a maximum) or from decreasing to increasing (indicating a minimum).