Find two positive numbers whose sum is 15 and | vedclass

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

Question

(A) Let the two positive numbers be $x$ and $y$. Given that $x + y = 15$,so $y = 15 - x$.Let $S$ be the sum of the squares of the numbers,so $S = x^2

Vedclass · Accepted Answer

$7.5, 7.5$

Vedclass · Answer

(A) Let the two positive numbers be $x$ and $y$. Given that $x + y = 15$,so $y = 15 - x$.Let $S$ be the sum of the squares of the numbers,so $S = x^2 + y^2$.Substituting $y = 15 - x$,we get $S(x) = x^2 + (15 - x)^2 = x^2 + 225 - 30x + x^2 = 2x^2 - 30x + 225$.To find the minimum,we find the first derivative $S'(x) = 4x - 30$.Setting $S'(x) = 0$,we get $4x = 30$,which implies $x = \frac{30}{4} = 7.5$.Now,find the second derivative $S''(x) = 4$.Since $S''(7.5) = 4 > 0$,the function $S(x)$ has a local minimum at $x = 7.5$.Thus,the two numbers are $x = 7.5$ and $y = 15 - 7.5 = 7.5$.

Find two positive numbers whose sum is $15$ and the sum of whose squares is minimum.

Similar Questions

The maximum value of $(1/x)^x$ is -

If $A = \{x \in R : \frac{\pi}{4} \leq x \leq \frac{\pi}{3}\}$ and $f(x) = \sin x - x$,then $f(A)$ is equal to

The maximum volume (in $m^3$) of the right circular cone having slant height $3 \, m$ is

The minimum value of $2x^2+x-1$ is:

If $xy = c^2$,what is the minimum value of $ax + by$ (where $a > 0, b > 0$)?

Vedclass Test Series

Exam Paper Generator

Online Exam Module