question 2 of 10, step 1 of 2 consider the function: f(x) = (81x² + 49)/x step 1 of 2: find the critical…

question 2 of 10, step 1 of 2 consider the function: f(x) = (81x² + 49)/x step 1 of 2: find the critical values of the function. separate multiple answers with commas. answer how to enter your answer (opens in new window) selecting a radio button will replace the entered answer value(s) with the radio button value. if the radio button is not selected, the entered answer is used. x = none

question 2 of 10, step 1 of 2 consider the function: f(x) = (81x² + 49)/x step 1 of 2: find the critical values of the function. separate multiple answers with commas. answer how to enter your answer (opens in new window) selecting a radio button will replace the entered answer value(s) with the radio button value. if the radio button is not selected, the entered answer is used. x = none

Answer

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{81x^{2}+49}{x}$ as $f(x)=81x + \frac{49}{x}=81x+49x^{-1}$.

Step2: Find the derivative

Using the power - rule $(x^n)'=nx^{n - 1}$, the derivative $f'(x)$ is $f'(x)=81-49x^{-2}=81-\frac{49}{x^{2}}$.

Step3: Set the derivative equal to zero

Set $f'(x) = 0$, so $81-\frac{49}{x^{2}}=0$. Then $\frac{49}{x^{2}}=81$. Cross - multiply to get $49 = 81x^{2}$.

Step4: Solve for x

$x^{2}=\frac{49}{81}$, so $x=\pm\frac{7}{9}$.

Answer:

$x = \frac{7}{9},-\frac{7}{9}$