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Solution for The Number Which Exceeds Its Square by the Greatest Possible Quantity is (A) 1 2 (B) 1 4 (C) 3 4 (D) None of These - CBSE (Commerce) Class 12 - Mathematics

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Question

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

(a) \[\frac{1}{2}\]

(b) \[\frac{1}{4}\]

(c) \[\frac{3}{4}\]

(d) none of these

Solution

\[(a) \frac{1}{2}\]

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

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

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

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

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

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

\[ \Rightarrow 2x = 1\]

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

\[\text { Now }, \]

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

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

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

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Solution The Number Which Exceeds Its Square by the Greatest Possible Quantity is (A) 1 2 (B) 1 4 (C) 3 4 (D) None of These Concept: Graph of Maxima and Minima.
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