Final answer:
To find local minima and maxima, analyze critical points using the first and second derivative tests.
Step-by-step explanation:
One way to find local minima and maxima is by analyzing critical points, which are the points where the derivative of a function is zero or undefined. These points can be found by taking the derivative of the function and setting it equal to zero. The values of x that satisfy this equation are the critical points. To determine if these critical points are local minima or maxima, you can use the first and second derivative tests. The first derivative test involves evaluating the derivative on either side of the critical point to determine if the function is increasing or decreasing. If the function changes from increasing to decreasing, the critical point is a local maximum. If the function changes from decreasing to increasing, the critical point is a local minimum. The second derivative test involves evaluating the second derivative at the critical point. If the second derivative is positive, the critical point is a local minimum. If the second derivative is negative, the critical point is a local maximum. If the second derivative is zero or undefined, the test is inconclusive.