The sum of two positive numbers is given. If | vedclass

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

Question

(A) Let $x$ and $y$ be two positive numbers. Given that their sum is constant,$x + y = a$. Let $z = x^3 + y^3$. Substituting $y = a - x$,we get $z = x

Vedclass · Accepted Answer

they are equal

Vedclass · Answer

(A) Let $x$ and $y$ be two positive numbers. Given that their sum is constant,$x + y = a$. Let $z = x^3 + y^3$. Substituting $y = a - x$,we get $z = x^3 + (a - x)^3$. Differentiating with respect to $x$: $\frac{dz}{dx} = 3x^2 - 3(a - x)^2 = 3(x^2 - (a^2 - 2ax + x^2)) = 3(2ax - a^2) = 3a(2x - a)$. Setting $\frac{dz}{dx} = 0$,we find $2x - a = 0$,which implies $x = \frac{a}{2}$. Since $y = a - x$,we get $y = a - \frac{a}{2} = \frac{a}{2}$. Checking the second derivative: $\frac{d^2z}{dx^2} = 6a$. Since $a > 0$,$\frac{d^2z}{dx^2} > 0$,confirming a minimum at $x = \frac{a}{2}$. Thus,$x = y$,meaning the numbers are equal.

The sum of two positive numbers is given. If the sum of their cubes is minimum,then

Similar Questions

The minimum value of $\alpha$ for which the equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=\alpha$ has at least one solution in $\left(0, \frac{\pi}{2}\right)$ is ..........

The minimum value of $4e^{2x} + 9e^{-2x}$ is:

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?

Find the maximum and minimum values for the function given by $g(x)=-|x+1|+3$.

An enemy fighter jet is flying along the curve given by $y = x^2 + 2$. $A$ soldier is placed at $(3, 2)$ and wants to shoot down the jet when it is nearest to him. The nearest distance is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module