How can I use derivatives to find the minima of a function?

How can I use derivatives to find the minima of a function?

To find the minima of a function, you can use the first and second derivative tests. Would you like a step-by-step explanation?

Yes, please explain the first derivative test.

The first derivative test involves finding the critical points by setting the first derivative of the function to zero. These points are where the function's slope changes.

What do I do after finding the critical points?

After finding the critical points, you evaluate the second derivative at those points. If it's positive, you have a local minimum.

Can you give me an example of this process?

Sure! Let's say we have the function f(x) = x^2. The first derivative is f'(x) = 2x. Setting it to zero gives x = 0, and the second derivative f''(x) = 2 is positive, indicating a minimum at x = 0.