# Calc2.49

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Find all local minimums and maximums of the function $f(x)={\frac {x^{2}-1}{x}}$.
All local minimums and maximums will be at critical points. A critical point is a point where the function is defined and the derivative is either $0$ or undefined.
$f'(x)={\frac {x(2x)-1(x^{2}-1)}{x^{2}}}={\frac {x^{2}+1}{x^{2}}}\,$
A fraction is equal to $0$ only when the numerator is equal to $0$. However, $x^{2}+1$ is always positive so the derivative is never equal to $0$. It is undefined when $x=0$ but so is the original function. Therefore there are no critical points and no local minimums or maximums.