Finding Relative Extrema Using Derivatives

In calculus, finding the relative extrema of a function is a fundamental concept. Relative extrema, also known as local extrema, are points on a function's graph where the function reaches a maximum or minimum value within a specific neighborhood. These points are crucial for understanding the behavior of a function and can be used to solve optimization problems.

Understanding Relative Extrema

Imagine a roller coaster ride. As you go up and down the hills, you experience points where you reach the highest or lowest points within a section of the track. These are analogous to relative extrema in a function. A relative maximum is like the top of a hill, while a relative minimum is like the bottom of a valley.

Using Derivatives to Find Relative Extrema

The key to finding relative extrema lies in the concept of derivatives. The derivative of a function gives us the slope of the tangent line at any point on the function's graph.

Here's how derivatives help us find relative extrema:

1. **Find the Critical Points:** Set the derivative of the function equal to zero and solve for the values of x. These values are called critical points. At these points, the tangent line is horizontal, indicating a potential maximum or minimum.
2. **Identify the Nature of the Critical Points:** Use the second derivative test to determine whether each critical point corresponds to a relative maximum, a relative minimum, or neither. The second derivative tells us the concavity of the function at the critical point. If the second derivative is positive, the function is concave up (like a smile), indicating a relative minimum. If the second derivative is negative, the function is concave down (like a frown), indicating a relative maximum.

Example: Finding Relative Extrema

Let's consider the function f(x) = x³ - 3x² + 2. We'll find its relative extrema using the steps outlined above.

1. **Find the Derivative:** The derivative of f(x) is f'(x) = 3x² - 6x.
2. **Find the Critical Points:** Set f'(x) = 0 and solve for x:
   ```
   3x² - 6x = 0
   3x(x - 2) = 0
   x = 0 or x = 2
   ```
   Therefore, the critical points are x = 0 and x = 2.
3. **Use the Second Derivative Test:** Find the second derivative of f(x): f''(x) = 6x - 6.
   * At x = 0, f''(0) = -6, which is negative. This indicates a relative maximum at x = 0.
   * At x = 2, f''(2) = 6, which is positive. This indicates a relative minimum at x = 2.

To find the corresponding y-values for the relative extrema, substitute the critical points back into the original function f(x):

• f(0) = 0³ - 3(0)² + 2 = 2. The relative maximum is at (0, 2).
• f(2) = 2³ - 3(2)² + 2 = -2. The relative minimum is at (2, -2).

Conclusion

Finding relative extrema using derivatives is a powerful tool in calculus. It allows us to analyze the behavior of functions and identify points where they reach maximum or minimum values within a specific interval. This knowledge has applications in various fields, including optimization problems, economics, and physics.