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

Answer 1

\[\frac{1}{2}\]

 

\[\text { Let the required number be x . Then, } \]

\[f\left( x \right) = x - x^2 \]

\[ \Rightarrow f'\left( x \right) = 1 - 2x\]

\[\text { For a local maxima or a local minima, we must have } \]

\[f'\left( x \right) = 0\]

\[ \Rightarrow 1 - 2x = 0\]

\[ \Rightarrow 2x = 1\]

\[ \Rightarrow x = \frac{1}{2}\]

\[\text { Now }, \]

\[f''\left( x \right) = - 2 < 0\]

\[\text { So, } x = \frac{1}{2}\text {  is a local maxima }. \]

\[\text { Hence, the required number is } \frac{1}{2} . \]