`g(t) = tsqrt(4 - t), t < 3` Find the critical numbers of the function.

Textbook Question

Chapter 3, 3.1 - Problem 13 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to evaluate the critical numbers of the function and for this reason, you must differentiate the function with respect to t, using the product and chain rules, such that:

`g'(t) = (t*sqrt(4 - t))'`

`g'(t) = t'*sqrt(4 - t) + t*(sqrt(4 - t))'`

`g'(t) = sqrt(4 - t) + t*((4-t)')/(2sqrt(4 - t))`

`g'(t) = sqrt(4 - t) + (-t)/(2sqrt(4 - t))`

You need to solve for t the equation g'(t) = 0:

`sqrt(4 - t) + (-t)/(2sqrt(4 - t)) = 0`

`2(4 - t) - t = 0 => 8 - 2t - t = 0 => 3t = 8 => t = 8/3`

Hence, evaluating the critical values of the given function, yields `t = 8/3.`

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