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.


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