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Maths-

General

Easy

Question

If $x y equals c to the power of 2 end exponent$ , then minimum value of $a x plus b y$ is

$c \sqrt{a b}$
$2 c \sqrt{a b}$
$- c \sqrt{a b}$
$- 2 c \sqrt{a b}$

The correct answer is: $2 c square root of a b end root$

Given curve is
$x y equals c to the power of 2 end exponent blank rightwards double arrow y equals fraction numerator c to the power of 2 end exponent over denominator x to the power of 2 end exponent end fraction$

Let $f open parentheses x close parentheses equals a x plus b y equals a x plus fraction numerator b c to the power of 2 end exponent over denominator x to the power of 2 end exponent end fraction$

On differentiating w.r.t. $x$ , we get

$f to the power of ´ end exponent open parentheses x close parentheses equals a minus fraction numerator b c to the power of 2 end exponent over denominator x to the power of 2 end exponent end fraction$

For a maximum or minima, put $f to the power of ´ end exponent open parentheses x close parentheses equals 0$

$rightwards double arrow blank a x to the power of 2 end exponent minus b c to the power of 2 end exponent equals 0$

$rightwards double arrow blank x to the power of 2 end exponent equals fraction numerator b c to the power of 2 end exponent over denominator a end fraction blank rightwards double arrow blank x equals plus-or-minus c square root of fraction numerator b over denominator a end fraction end root$

Again on differentiating, we get $f to the power of ´ ´ end exponent open parentheses x close parentheses equals fraction numerator 2 b c to the power of 2 end exponent over denominator x to the power of 3 end exponent end fraction$

At6 $x equals c square root of fraction numerator b over denominator a end fraction end root comma blank f to the power of ´ ´ end exponent open parentheses x close parentheses greater than 0$

$therefore blank f open parentheses x close parentheses$ is minimum at $x equals c square root of fraction numerator b over denominator a end fraction end root$

The minimum value at $x equals c square root of fraction numerator b over denominator a end fraction end root$ is

$therefore blank f open parentheses c square root of fraction numerator b over denominator a end fraction end root close parentheses equals a. c square root of fraction numerator b over denominator a end fraction end root plus fraction numerator b c to the power of 2 end exponent over denominator c end fraction. blank square root of fraction numerator a over denominator b end fraction end root$

$equals fraction numerator a b c plus a b c over denominator square root of a b end root end fraction equals fraction numerator 2 a b c over denominator square root of a b end root end fraction equals 2 c square root of a b end root$

Related Questions to study

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