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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 (Arts) Class 12 - Mathematics

Question

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

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{3}{4}$

• none of these

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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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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