If x, y are two positive integers such that x | vedclass

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

Question

(C) Given $x+y=20$. We want to maximize $f(x, y) = x^3 y$. Using the $AM$-$GM$ inequality for the terms $\frac{x}{3}, \frac{x}{3}, \frac{x}{3}, y$: $\

Vedclass · Accepted Answer

$\frac{\alpha}{\beta}$

Vedclass · Answer

(C) Given $x+y=20$. We want to maximize $f(x, y) = x^3 y$. Using the $AM$-$GM$ inequality for the terms $\frac{x}{3}, \frac{x}{3}, \frac{x}{3}, y$: $\frac{\frac{x}{3} + \frac{x}{3} + \frac{x}{3} + y}{4} \geq \sqrt[4]{\left(\frac{x}{3}ight)^3 y}$ $\frac{x+y}{4} \geq \sqrt[4]{\frac{x^3 y}{27}}$ $\frac{20}{4} \geq \sqrt[4]{\frac{x^3 y}{27}} \Rightarrow 5 \geq \sqrt[4]{\frac{x^3 y}{27}}$ $5^4 \geq \frac{x^3 y}{27} \Rightarrow x^3 y \leq 27 	imes 625 = 16875$. So,$k = 16875$. The maximum occurs when $\frac{x}{3} = y$. Since $x+y=20$,we have $3y+y=20$ $\Rightarrow 4y=20$ $\Rightarrow y=5=\beta$ and $x=15=\alpha$. Now,$\frac{k}{\alpha^2 \beta^2} = \frac{27 	imes 625}{15^2 	imes 5^2} = \frac{27 	imes 625}{225 	imes 25} = \frac{16875}{5625} = 3$. Since $\frac{\alpha}{\beta} = \frac{15}{5} = 3$,the correct option is $\frac{\alpha}{\beta}$.

If $x, y$ are two positive integers such that $x+y=20$ and the maximum value of $x^3 y$ is $k$ at $x=\alpha, y=\beta$,then $\frac{k}{\alpha^2 \beta^2} =$

Similar Questions

If $g(x) = 2f(2x^3 - 3x^2) + f(6x^2 - 4x^3 - 3)$,$\forall x \in R$ and $f''(x) > 0$,$\forall x \in R$,then $g'(x) > 0$ for $x$ belonging to

The curve $y(x) = ax^{3} + bx^{2} + cx + 5$ touches the $x$-axis at the point $P(-2, 0)$ and cuts the $y$-axis at the point $Q$,where the derivative $y'(0) = 3$. Find the local maximum value of $y(x)$.

Find the point where $f(x) = 2\sin x + \cos 2x$ has a maximum value in the interval $0 \leq x \leq 2\pi$.

Let $f(x) = x^2 + \frac{1}{x^2}$ and $g(x) = x - \frac{1}{x}$,$x \in R - \{-1, 1, 0\}$. If $h(x) = \frac{f(x)}{g(x)}$,then the local minimum value of $h(x)$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module