Find all critical points of the function f(t) = t - 18*sqrt(t + 1). (Give the answers as a...

Question:

Find all critical points of the function {eq}f(t) = t - 18 \sqrt{t + 1} {/eq}.

(Give the answers as a comma-separated list.)

Critical Points:

The critical points of a function are those points where the derivative is zero or undefined. They are obtained by finding the derivative of the function, setting it to zero and solving, and then adding the singularities to the list of solutions.

Taking the derivative, we have {eq}f'(t) = 1 - \frac{9}{\sqrt{t + 1} } {/eq}

Setting to zero and solving, we have {eq}0= 1 - \frac{9}{\sqrt{t + 1} } \\ 1 = \frac{9}{\sqrt{t + 1} } \\ \sqrt{t + 1} = 9 \\ t+1 = 81 \\ t = 80 {/eq}

Also, the derivative has a singularity where the denominator of the fraction is zero, which occurs when {eq}0=\sqrt{t+1} 0 = t+1 -1 = t {/eq}

Thus, the critical points are: -1, 80