The minimum value of 2x + 3y, when xy = 6, is | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $f(x, y) = 2x + 3y$ subject to the constraint $xy = 6.$Since $xy = 6,$ we have $y = \frac{6}{x}.$Substituting this into the function,we get $f

Vedclass · Accepted Answer

$12$

Vedclass · Answer

(A) Let $f(x, y) = 2x + 3y$ subject to the constraint $xy = 6.$Since $xy = 6,$ we have $y = \frac{6}{x}.$Substituting this into the function,we get $f(x) = 2x + 3\left(\frac{6}{x}\right) = 2x + \frac{18}{x}.$To find the minimum,we find the derivative $f'(x) = 2 - \frac{18}{x^2}.$Setting $f'(x) = 0,$ we get $2 = \frac{18}{x^2},$ which implies $x^2 = 9,$ so $x = \pm 3.$Since $x$ and $y$ must be positive for a minimum value in this context (as $xy=6$),we take $x = 3.$Then $y = \frac{6}{3} = 2.$The minimum value is $f(3) = 2(3) + 3(2) = 6 + 6 = 12.$

The minimum value of $2x + 3y,$ when $xy = 6,$ is

Similar Questions

Let the cubic polynomial $f(x) = x^3 - px + q$ have three real roots,where $p > 0$ and $q > 0$. Which of the following is true?

The maximum value of $f(x) = (x + 1)^{\frac{1}{3}} - (x - 1)^{\frac{1}{3}}$ for $x \in [0, 1]$ is ....

Divide $10$ into two parts such that the sum of double of the first and the square of the second is minimum.

The function $f(x)=x^3+a x^2+b x+c$ with $a^2 \leq 3 b$ has:

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius $3$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module