consider the function: f(x) = (9x² - 100)/x step 2 of 2: use the first derivative test to find any local…

consider the function: f(x) = (9x² - 100)/x step 2 of 2: use the first derivative test to find any local extrema. enter any local extrema as an ordered pair. answer separate multiple answers with commas. 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. local maxima: no local maxima local minima: no local minima

consider the function: f(x) = (9x² - 100)/x step 2 of 2: use the first derivative test to find any local extrema. enter any local extrema as an ordered pair. answer separate multiple answers with commas. 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. local maxima: no local maxima local minima: no local minima

Answer

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{9x^{2}-100}{x}$ as $f(x)=9x - \frac{100}{x}=9x-100x^{-1}$.

Step2: Find the first - derivative

Using the power rule $(x^n)'=nx^{n - 1}$, we have $f'(x)=9 + 100x^{-2}=9+\frac{100}{x^{2}}$.

Step3: Find the critical points

Set $f'(x) = 0$. Then $9+\frac{100}{x^{2}}=0$. Rearranging gives $\frac{100}{x^{2}}=-9$, or $100=-9x^{2}$. Since $x^{2}\geq0$ for all real $x$, and $-9x^{2}\leq0$ for all real $x$, and $100>0$, there are no real - valued solutions for $x$ from $f'(x) = 0$. Also, the function $f(x)$ is undefined at $x = 0$, but the domain of $f(x)$ is $x\neq0$. Since there are no critical points in the domain of $f(x)$ (points where $f'(x)=0$ or $f'(x)$ is undefined within the domain of $f$), we can conclude the following.

Answer:

Local Maxima: No Local Maxima Local Minima: No Local Minima