If $x_1, x_2, x_3, x_4 > 0$ and $x_1+x_2+x_3+x_4 =2$ , then $P = (x_1 + x_2) (x_3 + x_4)$ is bounded between?

Question

If $x_1, x_2, x_3, x_4 > 0$ and $x_1+x_2+x_3+x_4 =2$, then $P = (x_1 + x_2) (x_3 + x_4)$ is bounded between,

A. 0 and 1

B. 1 and 2

C. 2 and 3

D. 3 and 4

How do you solve these kinds of questions? I usually put many values and check the results like putting all = 0.5 gives $P = 1$ etc but how to actually solve these questions in a general way.

Answer 1

Remark that your problem is equivalent to : let $x,y>0$ such that $x+y=2$, then $xy$ is between ...

• By taking $x\rightarrow 0$ and $y\rightarrow 2$ for example, you can see that the lower bound of $xy$ is $0$.
• For the upper bound, you have in general $xy\leq \frac{(x+y)^2}{4}$, so that $xy\leq 1$. Remark that this upper bound is achieved for $x=y=1$.