Finding Local Maximum and Minimum using Critical Values

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In this example problem, we find the derivative of the function then we identify critical values by determining when the derivative will be undefined but disregard this x-value as the original function is not defined there. We set the derivative equal to zero and solve the resulting equation to identify two critical values. We use the 2nd derivative test to determine if the critical numbers will result in the original graph being concave up or concave down, thus telling us if they will be a local maximum or local minimum. We finish by evaluating our critical values into the original function to find the local maximum and local minimum.