Find the critical numbers of the function. (Enter your answers as a
\[
\begin{array}{l}
F(x)=x^{4 / 5}(x-5)^{2} \\
x=\square
\end{array}
\]
Answer
Final Answer: The critical numbers of the function are \(\boxed{1.43}\) and \(\boxed{5.00}\).
Step 1 ：To find the critical numbers of the function, we need to find where the derivative of the function is equal to zero or undefined.
Step 2 ：The derivative of the function can be found using the product rule and the chain rule. The product rule states that the derivative of two functions multiplied together is the first function times the derivative of the second plus the second function times the derivative of the first. The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.
Step 3 ：Let's denote the function as \( F = x^{4/5}(x - 5)^2 \).
Step 4 ：The derivative of the function is \( F' = 0.8(x - 5)^2/x^{0.2} + x^{0.8}(2x - 10) \).
Step 5 ：The critical numbers of the function are the solutions to the equation \( F'(x) = 0 \).
Step 6 ：The critical numbers of the function are approximately 1.43 and 5.00.
Step 7 ：Final Answer: The critical numbers of the function are \(\boxed{1.43}\) and \(\boxed{5.00}\).
