Find all critical points of the function f(t)=t-8sqrt(t+3). (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.) critical points:

Question

Find all critical points of the function  f(t)=t-8sqrt(t+3). (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.) critical points:

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Find Derivative of f(t): To find the critical points of the function, we need to find the values of t where the derivative of f(t) is either zero or undefined. First, let's find the derivative of f(t) with respect to t. f(t) = t - 8sqrt(t + 3) Using the chain rule and the power rule, the derivative f'(t) is: f'(t) = 1 - (8/2)(1/sqrt(t + 3)) Simplify the derivative: f'(t) = 1 - 4/sqrt(t + 3)Find Critical Points: Now, we need to find the values of t where f'(t) is zero or undefined. Set the derivative equal to zero and solve for t: 1 - 4/sqrt(t + 3) = 0 4/sqrt(t + 3) = 1 Multiply both sides by sqrt(t + 3) to clear the denominator: 4 = sqrt(t + 3) Square both sides to eliminate the square root: 16 = t + 3 Subtract 3 from both sides: t = 13Check Derivative Undefined: We also need to check where the derivative is undefined. The derivative f'(t) is undefined when the denominator is zero: sqrt(t + 3) = 0 Square both sides: t + 3 = 0 t = -3 However, t = -3 is not in the domain of the original function f(t) because we cannot take the square root of a negative number in the real number system. Therefore, t = -3 is not a critical point.Final Critical Point: The only critical point of the function is t = 13, where the derivative is zero.

Find all critical points of the function $\newline$ $f(t)=t-8\sqrt{t+3}.$ $\newline$ (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. If the function does not have any critical points, enter DNE.) $\newline$ critical points:

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