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Find the critical numbers of the function F(x) = x^(4/5)(x - 2)².

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Final answer:

To find the critical numbers of the function F(x) = x^(4/5)(x - 2)², we need to find the values of x where the derivative of F(x) is equal to 0 or does not exist. The critical numbers of the function F(x) are x = 0 and x = 16/9.

Step-by-step explanation:

To find the critical numbers of the function F(x) = x^(4/5)(x - 2)², we need to find the values of x where the derivative of F(x) is equal to 0 or does not exist.

First, we find the derivative of F(x) using the product rule and the chain rule: F'(x) = (4/5)x^(-1/5)(x - 2)² + x^(4/5)(2(x - 2)).

Next, we set F'(x) equal to 0 and solve for x. After simplifying the equation, we get x = 0 and x = 16/9.

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