PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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# Solution for The Sum of Two Non-zero Numbers is 8, the Minimum Value of the Sum of the Reciprocals is (A) 1 4 (B) 1 2 (C) 1 8 (D) None of These - PUC Karnataka Science Class 12 - Mathematics

#### Question

The sum of two non-zero numbers is 8, the minimum value of the sum of the reciprocals is

(a) $\frac{1}{4}$

(b) $\frac{1}{2}$

(c) $\frac{1}{8}$

(d) none of these

#### Solution

$(b) \frac{1}{2}$

$\text { Let the two non - zero numbers be x and y . Then,}$

$x + y = 8$

$\Rightarrow y = 8 - x . . . \left( 1 \right)$

$\text { Now,}$

$f\left( x \right) = \frac{1}{x} + \frac{1}{y}$

$\Rightarrow f\left( x \right) = \frac{1}{x} + \frac{1}{8 - x} \left[ \text { From eq } . \left( 1 \right) \right]$

$\Rightarrow f'\left( x \right) = \frac{- 1}{x^2} + \frac{1}{\left( 8 - x \right)^2}$

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

$f'\left( x \right) = 0$

$\Rightarrow \frac{- 1}{x^2} + \frac{1}{\left( 8 - x \right)^2} = 0$

$\Rightarrow \frac{- \left( 8 - x \right)^2 + x^2}{\left( x \right)^2 \left( 8 - x \right)^2} = 0$

$\Rightarrow - 64 - x^2 + 16x + x^2 = 0$

$\Rightarrow 16x - 64 = 0$

$\Rightarrow x = 4$

$f''\left( x \right) = \frac{2}{x^3} - \frac{2}{\left( 8 - x \right)^3}$

$\Rightarrow f''\left( 4 \right) = \frac{2}{4^3} - \frac{2}{\left( 8 - 4 \right)^3}$

$\Rightarrow f''\left( 4 \right) = \frac{2}{64} - \frac{2}{64} = 0$

$\therefore \text { Minimum value }= \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$

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Solution The Sum of Two Non-zero Numbers is 8, the Minimum Value of the Sum of the Reciprocals is (A) 1 4 (B) 1 2 (C) 1 8 (D) None of These Concept: Graph of Maxima and Minima.
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