How do you find local maxima and minima using derivatives?

Finding Local Maxima and Minima with Derivatives

To determine where a function has local maxima or minima (collectively called extrema), you use calculus:

Steps using derivatives:

• Find critical points:
1. Compute the first derivative, f'(x).
2. Solve f'(x) = 0 or where f'(x) is undefined.
3. Classify each critical point:
4. Use the second derivative, f''(x), for the Second Derivative Test:
5. If f''(c) > 0, you have a local minimum at x = c.
6. If f''(c) < 0, you have a local maximum at x = c.
7. If f''(c) = 0, the test is inconclusive-try other methods (like the First Derivative Test).

Example:
Let f(x) = x^3 - 3x + 1

8. Compute f'(x) = 3x^2 - 3
9. Set f'(x) = 0:

3x^2 - 3 = 0 → x^2 = 1 → x = -1, 1

10. Find f''(x) = 6x
• At x = -1: f''(-1) = -6 (max)
• At x = 1: f''(1) = 6 (min)

Takeaway:

• Use f'(x) to find critical points, then check f''(x) to classify each.
• This method helps identify where a function turns up (minimum) or down (maximum).