How to Find and Classify Local Extrema of a Function: A Step-by-Step Guide

The concept of finding and classifying local extrema is a fundamental part of calculus and is widely used in various fields, including optimization problems, economics, and engineering. This article will guide you through the steps to determine the local extrema of the function ( f(x) 35x^2 - 2x^5 ) and classify them using the second derivative test.

Step 1: Identify the First Derivative

To begin, we first need to find the first derivative of the given function, ( f(x) ).

[ f(x) 35x^2 - 2x^5 ]

The derivative of ( f(x) ) is:

[ f'(x) frac{d}{dx}(35x^2 - 2x^5) 7 - 1^4 ]

Setting ( f'(x) ) to zero will give us the critical points:

[ 7 - 1^4 0 ]

This equation can be factored to:

[ 1(7 - x^3) 0 ]

Thus, the critical points are:

[ x 0, , x sqrt[3]{7} approx 1.913 ]

Step 2: Determine the Second Derivative

The next step is to find the second derivative of the function, ( f(x) ).

[ f''(x) frac{d}{dx}(7 - 1^4) 70 - 4^3 ]

Step 3: Apply the Second Derivative Test

The second derivative test is used to determine whether a critical point is a local minimum, a local maximum, or neither. We will apply this test to the critical points obtained in Step 1.

[ f''(x) 70 - 4^3 ]

Evaluate the second derivative at each critical point:

( x 0 ): ( f''(0) 70 - 40(0)^3 70 gt 0 ) ( x sqrt[3]{7} ): ( f''(sqrt[3]{7}) 70 - 40(sqrt[3]{7})^3 70 - 40 cdot 7 70 - 280 -210 lt 0 )

Conclusion and Classification

Based on the second derivative test, we can conclude the following:

Local Minimum: At ( x 0 ), since ( f''(0) 70 gt 0 ), the function has a local minimum at this point. Local Maximum: At ( x sqrt[3]{7} approx 1.913 ), since ( f''(sqrt[3]{7}) -210 lt 0 ), the function has a local maximum at this point.

This analysis provides a clear understanding of the behavior of the function ( f(x) 35x^2 - 2x^5 ) and its critical points. By following these steps, you can efficiently find and classify local extrema for any given function.

Further Reading and Resources

For a deeper understanding of local extrema and the second derivative test, consider exploring the following resources:

Online Calculus Tutorials: Websites like Khan Academy (Khan Academy) offer comprehensive tutorials on optimization and the second derivative test. Textbooks: Books such as Calculus: Early Transcendentals by James Stewart provide in-depth explanations and exercises on these topics. Interactive Graphs: Websites like WolframAlpha (WolframAlpha) enable you to visualize functions and their extrema.

By mastering these concepts, you will be well-equipped to tackle a wide range of problems in calculus and its applications.