Question

consider the function (f(x)=81x^{1/3}-3x) over the domain (0,140). a) list and classify any critical values: classified as: choose one and choose one b) at (x = 0), choose the correct classification: relative minimum

Explanation:

Step1: Find the derivative of the function

The function is $f(x)=81x^{\frac{1}{3}}-3x$. Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, we have $f^\prime(x)=81\times\frac{1}{3}x^{-\frac{2}{3}}-3=27x^{-\frac{2}{3}}-3=\frac{27}{x^{\frac{2}{3}}}-3$.

Step2: Set the derivative equal to zero to find critical points

Set $f^\prime(x) = 0$, so $\frac{27}{x^{\frac{2}{3}}}-3 = 0$. Add 3 to both sides: $\frac{27}{x^{\frac{2}{3}}}=3$. Cross - multiply to get $27 = 3x^{\frac{2}{3}}$. Then divide both sides by 3: $x^{\frac{2}{3}} = 9$. Raise both sides to the $\frac{3}{2}$ power. We get $x=\pm27$. But our domain is $[0,140]$, so we only consider $x = 27$. Also, the derivative is undefined when $x = 0$ (since $f^\prime(x)=\frac{27}{x^{\frac{2}{3}}}-3$ and division by zero occurs at $x = 0$). So the critical values in the domain $[0,140]$ are $x = 0$ and $x=27$.

Step3: Classify the critical points

We use the second - derivative test. First, find the second - derivative $f^{\prime\prime}(x)=27\times(-\frac{2}{3})x^{-\frac{5}{3}}=-18x^{-\frac{5}{3}}=-\frac{18}{x^{\frac{5}{3}}}$.
For $x = 0$, the second - derivative is undefined.
For $x = 27$, $f^{\prime\prime}(27)=-\frac{18}{27^{\frac{5}{3}}}=-\frac{18}{(3^3)^{\frac{5}{3}}}=-\frac{18}{3^5}=-\frac{18}{243}<0$. So $x = 27$ is a relative maximum.
At $x = 0$, we can look at the behavior of the function. As $x$ approaches 0 from the right, $f(x)=81x^{\frac{1}{3}}-3x$. $f(0)=0$. We can also consider the first - derivative. For $x>0$ and close to 0, $f^\prime(x)=\frac{27}{x^{\frac{2}{3}}}-3>0$. So the function is increasing on an interval starting at $x = 0$, and $x = 0$ is a relative minimum.

a) Critical values: $x = 0,x = 27$; Classification: $x = 0$ is a relative minimum, $x = 27$ is a relative maximum.
b) At $x = 0$, the classification is relative minimum.

Answer:

a) Critical values: $x = 0,x = 27$; Classification: $x = 0$ is a relative minimum, $x = 27$ is a relative maximum.
b) relative minimum