Differentiability

Step 1: Analyze the given function

The given function is: where is a constant, and represents a fractional power of .

Step 2: Differentiate the function

To find the maxima or minima of , we need to first find the derivative and set it equal to zero. The derivative of is:

Step 3: Find critical points

Set the derivative equal to zero to find critical points: This equation will never be equal to zero because is never zero for any . Therefore, the function does not have any critical points where the derivative is zero.

Step 4: Check the behavior at

The function involves a fractional power, so we need to check the behavior of near , where the term becomes zero. - For , we have: - As (approaching 3 from the right), increases as increases slightly from 3. - As (approaching 3 from the left), also increases. Therefore, is decreasing for both and , and the maximum occurs at .

The maximum value of occurs at , and the maximum value is .