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


f(x) = -2x⁴ + x² + 10

Verified step by step guidance

1. First, find the derivative of the function f(x) = -2x⁴ + x² + 10. The derivative, f'(x), will help us determine where the function is increasing or decreasing.
2. Calculate the derivative: f'(x) = d/dx(-2x⁴ + x² + 10). Use the power rule for differentiation: d/dx(xⁿ) = n*xⁿ⁻¹.
3. Apply the power rule: f'(x) = -8x³ + 2x.
4. Set the derivative f'(x) equal to zero to find the critical points: -8x³ + 2x = 0. Factor the equation: 2x(-4x² + 1) = 0.
5. Solve for x: 2x = 0 gives x = 0, and -4x² + 1 = 0 gives x = ±√(1/4). Use these critical points to test intervals around them to determine where f'(x) is positive (increasing) or negative (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 is increasing before and decreasing after the critical point.

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, while it is decreasing if the derivative is negative. Identifying these intervals helps in understanding the overall behavior and trends of the function.