Critical Points and Local Maximum

$y = x^{2/3}(2-x)$

Find: A. The critical values of x? B. The x coordinate of the local maximum

Critical points and local maximum problems require finding where the function's slope is zero or undefined, as these points could indicate a local maximum, minimum, or a saddle point. The general approach is to first take the derivative of the function, which gives you the rate of change (or slope) of the function at any point. The critical values are found by setting this derivative equal to zero or determining where it is undefined, and then solving for the corresponding values of x. These are the points where the function could potentially have a local maximum or minimum.

For the given function, after finding the critical values, you can determine which one corresponds to a local maximum by checking the behavior of the function around each critical value, using a sign chart or the second derivative test. The x-coordinate of the local maximum will be the point where the function changes from increasing to decreasing, indicating that the function reaches a peak at that value.

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