The two positive numbers with sum t,and the s | vedclass

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

Question

(C) Let the two positive numbers be $x$ and $y$. Given that $x + y = t$,so $y = t - x$. We want to minimize the sum of their squares,$S = x^2 + y^2$.

Vedclass · Accepted Answer

$\frac{t}{2}, \frac{t}{2}$

Vedclass · Answer

(C) Let the two positive numbers be $x$ and $y$. Given that $x + y = t$,so $y = t - x$. We want to minimize the sum of their squares,$S = x^2 + y^2$. Substituting $y = t - x$,we get $S(x) = x^2 + (t - x)^2 = x^2 + t^2 - 2tx + x^2 = 2x^2 - 2tx + t^2$. To find the minimum,we take the derivative with respect to $x$: $\frac{dS}{dx} = 4x - 2t$. Setting $\frac{dS}{dx} = 0$,we get $4x = 2t$,which implies $x = \frac{t}{2}$. Since $\frac{d^2S}{dx^2} = 4 > 0$,the function has a minimum at $x = \frac{t}{2}$. Then $y = t - \frac{t}{2} = \frac{t}{2}$. Thus,the two numbers are $\frac{t}{2}$ and $\frac{t}{2}$.

The two positive numbers with sum $t$,and the sum of their squares is minimum are

Similar Questions

Consider the region $R = \{(x, y) : x \leq y \leq 9 - \frac{11}{3} x^2, x \geq 0\}$. The area of the largest rectangle with sides parallel to the coordinate axes and inscribed in $R$ is:

Let $f(x) = \int\limits_0^x \frac{\sin t}{t} dt$ for $x > 0$. Then $f(x)$ has:

The minimum value of the function $f(x) = 3x^4 - 8x^3 + 12x^2 - 48x + 25$ on the interval $[0, 3]$ is equal to:

The number that exceeds its square by the greatest amount is

The maximum area of a rectangle that can be inscribed in a circle of radius $2 \text{ unit}$ is (in square unit)

Vedclass Test Series

Exam Paper Generator

Online Exam Module