Find the critical numbers of the function. (Enter your answers as a comma-separated list.) $$ g(t)=t \sqrt{5-t}, \quad t

Question

Find the critical numbers of the function. (Enter your answers as a comma-separated list.) $$ g(t)=t \sqrt{5-t}, \quad t<4 $$ $$ t=\square $$

UpStudy AI · Accepted Answer

Function by following steps:
- step0: Find the critical numbers:
  $$g\left(t\right)=t\sqrt{5-t}$$
- step1: Find the domain:
  $$g\left(t\right)=t\sqrt{5-t},t\leq 5$$
- step2: Find the derivative:
  $$g^{\prime}\left(t\right)=\frac{10-3t}{2\sqrt{5-t}}$$
- step3: Find the domain:
  $$g^{\prime}\left(t\right)=\frac{10-3t}{2\sqrt{5-t}},t<5$$
- step4: Substitute $$g^{\prime}\left(t\right)=0:$$
  $$0=\frac{10-3t}{2\sqrt{5-t}}$$
- step5: Swap the sides:
  $$\frac{10-3t}{2\sqrt{5-t}}=0$$
- step6: Cross multiply:
  $$10-3t=2\sqrt{5-t}\times 0$$
- step7: Simplify the equation:
  $$10-3t=0$$
- step8: Move the constant to the right side:
  $$-3t=0-10$$
- step9: Remove 0:
  $$-3t=-10$$
- step10: Change the signs:
  $$3t=10$$
- step11: Divide both sides:
  $$\frac{3t}{3}=\frac{10}{3}$$
- step12: Divide the numbers:
  $$t=\frac{10}{3}$$
- step13: Check if the solution is in the defined range:
  $$t=\frac{10}{3},t<5$$
- step14: Find the intersection:
  $$t=\frac{10}{3}$$
- step15: Check the solution:
  $$t=\frac{10}{3},t\leq 5$$
- step16: Find the critical numbers:
  $$\begin{align}&t=\frac{10}{3}\&t=5\end{align}$$
The critical numbers of the function $$ g(t) = t \sqrt{5 - t} $$ are:

$$
t = \frac{10}{3}, t = 5
$$

However, since the condition states $$ t < 4 $$, we only consider $$ t = \frac{10}{3} $$ as the valid critical number.

Thus, the answer is:

$$ t = \frac{10}{3} $$
$$ t = \frac{10}{3} $$

Find the critical numbers of the function. (Enter your answers as a comma-separated list.)

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