Question

Let positive real numbers x and y be such that 3x+4y =14. The maximum value of $x^3{y}^4$is

• A. $1056$
• B. $128$
• C. $216$
• D. $432$

Answer 1

Answer: (A)

$1056$

We have,

$3x+4y=14.......\left(1\right)$

$4y=14-3x$

$y=\frac{14-3x}{4}$

Let $S$ be the maximum

So,

$S=x^3{y}^4$

$S=x^3{\left(\frac{14-3x}{4}\right)}^4$

$S=\frac{1}{256}{x}^3{(14-3x)}^4$

Differentiating this equation with respect to x and we get,

$\frac{ds}{dx}=\frac{1}{256}\frac{d}{dx}{x}^3{(14-3x)}^4$

$\frac{ds}{dx}=\frac{1}{256}[x^3\frac{d}{dx}{(14-3x)}^4+{(14-3x)}^4\frac{d}{dx}{x}^3]$

$\frac{ds}{dx}=\frac{1}{256}[x^{3}4{(14-3x)}^3(-3)+3x^2{(14-3x)}^4]$

$\frac{ds}{dx}=\frac{3}{256}[x^2{(14-3x)}^4-4x^3{(14-3x)}^3].......\left(2\right)$

For maximum and minimum

$\frac{ds}{dx}=0$

$\frac{3}{256}[x^2{(14-3x)}^4-4x^3{(14-3x)}^3]=0$

$x^2{(14-3x)}^4-4x^3{(14-3x)}^3=0$

$x^2{(14-3x)}^4=4x^3{(14-3x)}^3$

$4x^3{(14-3x)}^3=x^2{(14-3x)}^4$

$4x=(14-3x)$

$4x+3x=14$

$7x=14$

$x=2$

Again differentiating equation (2) and we get,

$\frac{d^{2}s}{d{x}^2}=\frac{d}{dx}\left[\frac{3}{256}[x^2{(14-3x)}^4-4x^3{(14-3x)}^3]\right]$

$=\frac{3}{256}[x^2\frac{d}{dx}{(14-3x)}^4+{(14-3x)}^4\frac{d}{dx}{x}^2-4(x^3\frac{d}{dx}{(14-3x)}^3+{(14-3x)}^3\frac{d}{dx}{x}^3)]$

$=\frac{3}{256}[x^{2}4{(14-3x)}^3(-3)+{(14-3x)}^{4}2x-4(x^{3}3{(14-3x)}^2(-3)+{(14-3x)}^{3}3x^2)]$

$=\frac{3}{256}[-12x^2{(14-3x)}^3+2x{(14-3x)}^4+36x^3{(14-3x)}^2-12x^2{(14-3x)}^3]$

$=\frac{3}{256}[2x{(14-3x)}^4-24x^2{(14-3x)}^3+36x^3{(14-3x)}^2]$

Put $x=2$ and we get,

$\frac{d^{2}s}{d{x}^2}=\frac{3}{256}[2x{(14-3x)}^4-24x^2{(14-3x)}^3+36x^3{(14-3x)}^2]$

$\frac{d^{2}s}{d{x}^2}=\frac{3}{256}[2\left(2\right){(14-3\left(2\right))}^4-24{\left(2\right)}^2{(14-3\left(2\right))}^3+36{\left(2\right)}^3{(14-3\left(2\right))}^2]$

$\frac{d^{2}s}{d{x}^2}=\frac{3}{256}[4\times 8^4-24\times 6\times 8^3+36\times 8\times 8^2]$

$\frac{d^{2}s}{d{x}^2}=\frac{3\times 8^3}{256}[4\times 8-24\times 6+36\times 8]$

$\frac{d^{2}s}{d{x}^2}=3\times \frac{512}{256}[32-144+288]$

$\frac{d^{2}s}{d{x}^2}=3\times 2[320-144]$

$\frac{d^{2}s}{d{x}^2}=6\times 176$

$\frac{d^{2}s}{d{x}^2}=1056$

Hence, this is the answer.