Hi, I'm having trouble with this inequality question:
Prove that, if a,b,c and d are any four positive numbers, then ab+cd <= sqr((a^2+c^2)(b^2+d^2))
I worked backwards and got that (a^2+c^2)(b^2+d^2) >=4abcd, and I need to prove that (ab+cd)^2<=4abcd. I don't know how to manipulate the LHS to do that, and also is this the right method?
Also, just generally: what's the best way to approach these kinds of inequality questions?
What are some useful inequality identities to know eg. a^2+b^2>=2ab or are you expected to derive them from the beginning?
I really appreciate you answering my questions!
When you write out your answer, you must start from the beginning, which is either the basic fact that \( (a-b)^2 \geq 0\), or some equation they've given to you. Of course, if there's a part i) or something, you can assume the result of that for part ii).
Working backwards is good to help determine how you would write it up in a forwards direction. But note that your final solution needs to be written the correct way no matter what.
\[ \text{With your problem, }\textbf{further}\text{ working backwards from}\\ (ab+cd)^2 \leq 4abcd\text{ will have provided the path.} \]
\begin{align*}(ab-cd)^2 &\geq 0\\ a^2b^2 - 2abcd + c^2d^2 &\geq 0\\ a^2b^2 + 2abcd + c^2d^2 &\geq 4abcd\\ (ab+cd)^2 &\geq 4abcd \end{align*}
Alternatively, if you've proven \( (x+y)^2 \geq 4xy\) using \( (x-y)^2 \geq 0\) from an earlier part, you can just sub \(x=ab\) and \(y=cd\) without regurgitating the same proof from scratch.
HOWEVER:
Later, when I worked backwards on what you were originally trying to prove, I obtained this.
\begin{align*}ab+cd&\leq \sqrt{(a^2+b^2)(c^2+d^2)}\\ a^2b^2+2abcd+c^2d^2 &\leq a^2b^2+a^2d^2+c^2b^2+c^2d^2\\ a^2d^2-2abcd+c^2b^2 &\geq 0\\ (ad-bc)^2 &\geq 0 \end{align*}
So to prove that, I would've written
that proof out backwards.
\begin{align*}(ad-bc)^2 &\geq 0\\ a^2d^2-2abcd + b^2c^2 &\geq 0\\ a^2b^2+a^2d^2+c^2b^2+c^2d^2 &\geq a^2b^2+2abcd+c^2d^2\\ (a^2+c^2)(b^2+d^2) &\geq (ab+cd)^2\\ \therefore ab+cd &\leq \sqrt{(a^2+c^2)(b^2+d^2)} \end{align*}
\[ \text{taking positive roots, since }ab+cd > 0\text{ from the given information.} \]
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