Determine the critical numbers, if any, of the function f on the interval [1,8]. f(x)=x^(2)sqrt(8-x) Give your answer as a comma-separated list. Express numbers in exact form. If the function does not have any critical numbers, enter DNE.

Question

Determine the critical numbers, if any, of the function  f on the interval  [1,8].  f(x)=x^(2)sqrt(8-x) Give your answer as a comma-separated list. Express numbers in exact form. If the function does not have any critical numbers, enter DNE.

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Identify Critical Numbers: Identify the critical numbers of the function f(x) = x^2 * sqrt(8-x). Critical numbers occur where the derivative f'(x) is zero or undefined.Calculate Derivative: Calculate the derivative of f(x) = x^2 * sqrt(8-x). Use the product rule and the chain rule. f'(x) = d/dx [x^2] * sqrt(8-x) + x^2 * d/dx [sqrt(8-x)] f'(x) = 2x * sqrt(8-x) + x^2 * (1/2) * (8-x)^(-1/2) * (-1) f'(x) = 2x * sqrt(8-x) - (x^2 / (2 * sqrt(8-x)))Simplify Derivative: Simplify the derivative to find where it is zero or undefined. f'(x) = (2x * (8-x) - x^2) / (2 * sqrt(8-x)) f'(x) = (16x - 2x^2 - x^2) / (2 * sqrt(8-x)) f'(x) = (16x - 3x^2) / (2 * sqrt(8-x))Find Zeroes: Set the numerator equal to zero to find where the derivative is zero. 16x - 3x^2 = 0 x(16 - 3x) = 0 x = 0 or x = 16/3Check Values: Check if the values found are in the interval [1,8]. x = 0 is not in the interval [1,8]. x = 16/3 is approximately 5.33, which is in the interval [1,8].Check Undefined Points: Check for points where the derivative is undefined by setting the denominator equal to zero. 2 * sqrt(8-x) = 0 sqrt(8-x) = 0 8 - x = 0 x = 8Verify Endpoint: Verify that x = 8 is in the interval [1,8]. Since 8 is the endpoint of the interval, it is considered a critical number if the function is defined there.Combine Critical Numbers: Combine the results from steps 5 and 7 to list all critical numbers in the interval [1,8]. The critical numbers are x = 16/3 and x = 8.

Determine the critical numbers, if any, of the function $f$ on the interval $[1,8]$ . $\newline$ $f(x)=x^{2}\sqrt{8-x}$ $\newline$ Give your answer as a comma-separated list. Express numbers in exact form. If the function does not have any critical numbers, enter DNE.

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Determine the critical numbers, if any, of the function $f$ on the interval $[1,8]$ . $\newline$ $f(x)=x^{2}\sqrt{8-x}$ $\newline$ Give your answer as a comma-separated list. Express numbers in exact form. If the function does not have any critical numbers, enter DNE.