The number which exceeds its square by the greatest possible quantity is _________________ .

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KollegeApply · Accepted Answer

$\frac{1}{2}$ \text { Let the required number be x . Then, } f\left( x \right) = x - x^2 \Rightarrow f'\left( x \right) = 1 - 2x \text { For a local maxima or a local minima, we must have } f'\left( x \right) = 0 \Rightarrow 1 - 2x = 0 \Rightarrow 2x = 1 \Rightarrow x = $\frac{1}{2}$ \text { Now }, f''\left( x \right) = - 2 < 0 \text { So, } x = $\frac{1}{2}$\text { is a local maxima }. \text { Hence, the required number is } $\frac{1}{2}$ .

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