Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x⁴ - 50x² on [-1, 5]

Verified step by step guidance

1. To find the critical points of the function \( g(x) = x^4 - 50x^2 \), first compute the derivative \( g'(x) \).
2. The derivative is \( g'(x) = 4x^3 - 100x \). Set \( g'(x) = 0 \) to find the critical points.
3. Solve the equation \( 4x^3 - 100x = 0 \). Factor out the common term: \( 4x(x^2 - 25) = 0 \).
4. This gives \( x = 0 \) or \( x^2 - 25 = 0 \). Solving \( x^2 - 25 = 0 \) gives \( x = 5 \) or \( x = -5 \).
5. Evaluate \( g(x) \) at the critical points \( x = 0, 5, -5 \) and at the endpoints of the interval \( x = -1 \) and \( x = 5 \) to determine the absolute maximum and minimum values.

Key Concepts

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima and minima, as they represent potential locations where the function's behavior changes. To find critical points, one must first compute the derivative of the function and solve for the values of x that satisfy the condition.

Absolute Maximum and Minimum

The absolute maximum and minimum values of a function on a closed interval are the highest and lowest values that the function attains within that interval. To determine these values, one must evaluate the function at its critical points and at the endpoints of the interval. The largest and smallest of these values will be the absolute maximum and minimum, respectively.

Endpoints of an Interval

Endpoints of an interval are the boundary values that define the limits of the interval in which a function is analyzed. In the context of finding absolute extrema, it is crucial to evaluate the function at these endpoints, as they can potentially yield the highest or lowest values alongside the critical points. For the function g(x) = x⁴ - 50x² on [-1, 5], the endpoints are x = -1 and x = 5.