If a, b, c, d are positive real numbers such | vedclass

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

Question

(A) Given $a, b, c, d > 0$ and $a + b + c + d = 2$.Let $x = a + b$ and $y = c + d$. Then $x + y = 2$.We want to find the range of $M = xy$.By the Arit

Vedclass · Accepted Answer

$0 < M \le 1$

Vedclass · Answer

(A) Given $a, b, c, d > 0$ and $a + b + c + d = 2$.Let $x = a + b$ and $y = c + d$. Then $x + y = 2$.We want to find the range of $M = xy$.By the Arithmetic Mean-Geometric Mean ($AM$-$GM$) inequality,we know that for positive real numbers $x$ and $y$,$\frac{x + y}{2} \ge \sqrt{xy}$.Substituting the values,we get $\frac{2}{2} \ge \sqrt{M}$,which implies $1 \ge \sqrt{M}$.Squaring both sides,we get $M \le 1$.Since $a, b, c, d$ are positive,$x = a + b > 0$ and $y = c + d > 0$,so $M = xy > 0$.Therefore,the relation satisfied is $0 < M \le 1$.

If $a, b, c, d$ are positive real numbers such that $a + b + c + d = 2$,then $M = (a + b)(c + d)$ satisfies the relation

Similar Questions

The product of three geometric means between $4$ and $\frac{1}{4}$ will be

If $a, b, c$ are in $A$.$P$.,then the straight line $ax + by + c = 0$ will always pass through the point

The sum of all two-digit numbers which,when divided by $4$,yield unity as a remainder is:

Let $a, b, c, d \in R^+$ such that $256 abcd \geq (a+b+c+d)^4$ and $3a + b + 2c + 5d = 11$. Then,the value of $a^3 + b + c^2 + 5d$ is equal to:

If the sum of the first $n$ terms of two arithmetic progressions $3, 7, 11, 15, \ldots$ and $30, 33, 36, 39, \ldots$ are equal,then find the value of $n$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module