Question

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

• A. 9
• B. 12
• C. 8
• D. 6

Answer 1

Answer: (B)

Let $f(x)=2x+3y$
$f(x)=2x+\frac{18}{x}$
$(\because xy=6$ given $)$
On differentiating, we get
$f^{\prime }x=2-\frac{18}{x^2}$
Put $f^{\prime }(x)=0$ for maximum or minima.
$\Rightarrow 0=2-\frac{18}{x^2}$
$\Rightarrow x=\pm 3$
And $f^{\prime \prime }x=\frac{36}{x^3}$
$\Rightarrow f^{\prime \prime }3=\frac{36}{3^3}>0$
$\therefore$ At $x=3$, If $x$ is minimum.
The minimum value is
$f(3)=2(3)+3(2)=12$