Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = -12x⁵ + 75x⁴ - 80x³

Verified step by step guidance

1. To determine where the function f(x) = -12x⁵ + 75x⁴ - 80x³ is increasing or decreasing, we first need to find its derivative, f'(x). This will help us identify the critical points and analyze the behavior of the function.
2. Calculate the derivative f'(x) using the power rule. The derivative of f(x) = -12x⁵ + 75x⁴ - 80x³ is f'(x) = -60x⁴ + 300x³ - 240x².
3. Find the critical points by setting the derivative f'(x) equal to zero and solving for x. This involves solving the equation -60x⁴ + 300x³ - 240x² = 0.
4. Factor the equation -60x⁴ + 300x³ - 240x² = 0 to find the values of x that make the derivative zero. This can be done by factoring out the greatest common factor and solving the resulting polynomial equation.
5. Once the critical points are found, use a sign chart or test values in the intervals determined by these critical points to determine where f'(x) is positive (indicating f is increasing) and where f'(x) is negative (indicating f is decreasing).

Key Concepts

Critical Points

Critical points are values of x where the derivative of a function is either zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. To find critical points, we first compute the derivative of the function and set it equal to zero, solving for x.

First Derivative Test

The First Derivative Test is a method used to determine the behavior of a function at its critical points. By analyzing the sign of the derivative before and after each critical point, we can conclude whether the function is increasing or decreasing in the intervals defined by these points. If the derivative changes from positive to negative, the function has a local maximum; if it changes from negative to positive, it has a local minimum.

Intervals of Increase and Decrease

Intervals of increase and decrease refer to the ranges of x-values where a function is respectively rising or falling. A function is increasing on an interval if its derivative is positive throughout that interval, and it is decreasing if the derivative is negative. Identifying these intervals helps in understanding the overall behavior of the function and is crucial for graphing and optimization problems.