question 3 of 14 >\nfind all critical points of the function (f(t)=t - 2sqrt{t}+9).\n(use symbolic notation…

Explanation:

Step1: Rewrite the function

Rewrite $f(t)=t - 2\sqrt{t}+9$ as $f(t)=t - 2t^{\frac{1}{2}}+9$.

Step2: Find the derivative

Using the power - rule $\frac{d}{dt}(t^n)=nt^{n - 1}$, we have $f^\prime(t)=\frac{d}{dt}(t)-2\frac{d}{dt}(t^{\frac{1}{2}})+\frac{d}{dt}(9)$. So $f^\prime(t)=1 - 2\times\frac{1}{2}t^{-\frac{1}{2}}+0=1-\frac{1}{\sqrt{t}}$.

Step3: Set the derivative equal to zero

Set $f^\prime(t) = 0$, then $1-\frac{1}{\sqrt{t}}=0$. Add $\frac{1}{\sqrt{t}}$ to both sides: $1=\frac{1}{\sqrt{t}}$. Cross - multiply to get $\sqrt{t}=1$. Square both sides, and we find $t = 1$.

Step4: Check where the derivative is undefined

The derivative $f^\prime(t)=1-\frac{1}{\sqrt{t}}$ is undefined when $t = 0$ since we have $\frac{1}{\sqrt{t}}$ and the square root of zero in the denominator is not allowed in the real - number system. But the domain of the original function $f(t)=t - 2\sqrt{t}+9$ is $t\gt0$ (because of the square root $\sqrt{t}$). So we only consider the value from setting the derivative equal to zero.

Answer:

$1$