does the function y= (x-1)/x^2 have a maximum or a minimum because when i conduct a derivative test there signs change from positive to negative at x=2, but when i graph the function on a graphing calculator there doesn't seem to be a maximum
does the function y= (x-1)/x^2 have a maximum or a minimum because when i conduct a derivative test there signs change from positive to negative at x=2, but when i graph the function on a graphing calculator there doesn't seem to be a maximum
The function does have a maximum, and we may calculate the derivative as follows:
$\displaystyle \begin{aligned}
\frac{d}{dx}\frac{x-1}{x^2}&=\frac{d}{dx}(x^{-1}-x^{-2})\\
&=\frac{d}{dx}(x^{-1})-\frac{d}{dx}(x^{-2})\\
&=-x^{-2}+2x^{-3}.
\end{aligned}
$
You are correct that the derivative changes sign at $\displaystyle x=2$, and as the function decreases after $\displaystyle x=2$, a maximum occurs here.