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Question
The number which exceeds its square by the greatest possible quantity is _________________ .
Options
\[\frac{1}{2}\]
\[\frac{1}{4}\]
\[\frac{3}{4}\]
none of these
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Solution
\[\frac{1}{2}\]
\[\text { Let the required number be x . Then, } \]
\[f\left( x \right) = x - x^2 \]
\[ \Rightarrow f'\left( x \right) = 1 - 2x\]
\[\text { For a local maxima or a local minima, we must have } \]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 1 - 2x = 0\]
\[ \Rightarrow 2x = 1\]
\[ \Rightarrow x = \frac{1}{2}\]
\[\text { Now }, \]
\[f''\left( x \right) = - 2 < 0\]
\[\text { So, } x = \frac{1}{2}\text { is a local maxima }. \]
\[\text { Hence, the required number is } \frac{1}{2} . \]
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