Hi, I'm having trouble solving this inequality:
'Let $\displaystyle p$ and $\displaystyle q$ be positive real numbers'. Prove that:
$\displaystyle (p + 2)(q + 2)(p + 1) \geq 16pq$
Please help explain how to solve it to me.
Thanks, BL
Hi, I'm having trouble solving this inequality:
'Let $\displaystyle p$ and $\displaystyle q$ be positive real numbers'. Prove that:
$\displaystyle (p + 2)(q + 2)(p + 1) \geq 16pq$
Please help explain how to solve it to me.
Thanks, BL
Maybe the inequality should be
$\displaystyle (p+2)(q+2)(p+q)\geq 16pq$.
That can be proved as follows.
$\displaystyle (x - y)^2 \geq 0$
$\displaystyle x^2 + y^2 - 2xy \geq 0, $
$\displaystyle x^2 + y^2 \geq 2xy.$
Now let $\displaystyle x = \sqrt{p}$ and $\displaystyle y = \sqrt{q}$, then
$\displaystyle p + q \geq 2\sqrt{pq}.$
Similarly $\displaystyle q + r \geq 2\sqrt{qr}$ and $\displaystyle r + p \geq 2\sqrt{rp}$.
Multiplying these three inequalities and putting $\displaystyle r = 2$, leads to the stated result.