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Let X, Y Be Two Variables and X>0, Xy=1, Then Minimum Value of X+Y is (A) 1 (B) 2 (C) 2 1 2 (D) 3 1 3 - CBSE (Arts) Class 12 - Mathematics

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Question

Let x, y be two variables and x>0, xy=1, then minimum value of x+y is _______________ .

  • 1

  • 2

  • \[2\frac{1}{2}\]

  • \[3\frac{1}{3}\]

Solution

2

 

\[\text { Given }: xy = 1\]

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

\[f\left( x \right) = x + \frac{1}{x}\]

\[ \Rightarrow f'\left( x \right) = 1 - \frac{1}{x^2}\]

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

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

\[ \Rightarrow 1 - \frac{1}{x^2} = 0\]

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

\[ \Rightarrow x^2 = 1\]

\[ \Rightarrow x = \pm 1\]

\[ \Rightarrow x = 1 ..............\left( \text { Given }: x > 1 \right)\]

`rArry=1`

\[\text { Now,} \]

\[f''\left( x \right) = \frac{2}{x^3}\]

\[ \Rightarrow f''\left( 1 \right) = 2 > 0\]

\[\text { So, x = 1 is a local minima } . \]

\[ \therefore \text { Minimum value of } f\left( x \right) = f\left( 1 \right) = 1 + 1 = 2\]

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Solution Let X, Y Be Two Variables and X>0, Xy=1, Then Minimum Value of X+Y is (A) 1 (B) 2 (C) 2 1 2 (D) 3 1 3 Concept: Graph of Maxima and Minima.
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