First Derivative Test


a. Locate the critical points of f.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).


f(x) = 2x⁵ - 5x⁴ - 10x³ + 4 on [-2,4]

Verified step by step guidance

1. To find the critical points of the function \( f(x) = 2x^5 - 5x^4 - 10x^3 + 4 \), first compute the derivative \( f'(x) \). Use the power rule to differentiate each term: \( f'(x) = 10x^4 - 20x^3 - 30x^2 \).
2. Set the derivative \( f'(x) \) equal to zero to find the critical points: \( 10x^4 - 20x^3 - 30x^2 = 0 \). Factor the equation to solve for \( x \).
3. Factor out the greatest common factor: \( 10x^2(x^2 - 2x - 3) = 0 \). Solve \( 10x^2 = 0 \) to find \( x = 0 \). Then solve \( x^2 - 2x - 3 = 0 \) using the quadratic formula or factoring to find additional critical points.
4. Use the First Derivative Test to determine the nature of each critical point. Evaluate \( f'(x) \) on intervals around each critical point to determine if the function is increasing or decreasing, which will indicate local maxima or minima.
5. To find the absolute maximum and minimum values on the interval \([-2, 4]\), evaluate \( f(x) \) at the critical points and the endpoints \( x = -2 \) and \( x = 4 \). Compare these values to determine the absolute extrema.

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 f'(x) = 0.

First Derivative Test

The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can conclude that if the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it is a local minimum.

Absolute Maximum and Minimum

The absolute maximum and minimum values of a function on a closed interval are the highest and lowest values the function attains within that interval. To find 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.