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Maths-

General

Easy

Question

The real number $x$ when added to its inverse gives the minimum value of the sum at $x$ which is equal to

$- 2$
2
1
$- 1$

The correct answer is: 1

Let $A equals x plus fraction numerator 1 over denominator x end fraction blank rightwards double arrow fraction numerator d A over denominator d x end fraction equals 1 minus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction$
Put $fraction numerator d A over denominator d x end fraction equals 0$ for maxima or minima

$1 minus fraction numerator 1 over denominator x to the power of 2 end exponent end fraction equals 0 blank rightwards double arrow blank x equals negative 1 comma blank 1$

Again on differentiating w.r.t. $x$ , we get

$fraction numerator d to the power of 2 end exponent A over denominator d x to the power of 2 end exponent end fraction equals fraction numerator 2 over denominator x to the power of 3 end exponent end fraction rightwards double arrow blank open parentheses fraction numerator d to the power of 2 end exponent A over denominator d x to the power of 2 end exponent end fraction close parentheses subscript x equals 1 end subscript equals 2 greater than 0$

$therefore A$ is minimum at $x equals 1$

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