Let $a,b\in \mathbb{R}$ such that $ab=2$. Find the max value of $\frac{3}{2\left(a+b\right)^2}$ and $a, b$ where max is attained, without calculus.

Question

My thinking:

By Arithmetic and Geometric mean inequality (AGM):

$ab\le \left(\frac{a+b}{2}\right)^2$

We know $ab=2$

$\rightarrow$ $2\le \left(\frac{a+b}{2}\right)^2$

$\rightarrow$ $2\le \frac{a^2+b^2}{4}$

$\rightarrow$ $\frac{1}{2}\le \:\frac{1}{\frac{a^2+b^2}{4}}$

$\rightarrow$ $\frac{1}{2}\le \:\frac{4}{a^2+b^2}$

$\rightarrow$ $\frac{1}{2}\left(\frac{3}{a^2+b^2}\right)\le \:\frac{4}{a^2+b^2}\left(\frac{3}{a^2+b^2}\right)$

$\rightarrow$ $\frac{3}{2\left(a+b\right)^2}\le \frac{12}{\left(a+b\right)^4}$

Therefore the max value is $\frac{12}{\left(a+b\right)^4}$

Maximum value is attained when $a=b$:

So we can write:

$\rightarrow$ $\frac{3}{2\left(a\right)^2}=\frac{12}{\left(a\right)^4}$

$\rightarrow$ $3a^4=24a^2$

$\rightarrow$ $a=0, -2\sqrt{2},+2\sqrt{2}$

Therefore the max value of $a,b$ are $0,-2\sqrt{2},+2\sqrt{2}$

I'm not completely sure about my answer, if anyone could provide some feedback, that would be amazing! Thanks in advance.

EDIT:

$ab\le \frac{\left(a+b\right)^2}{2}\:\rightarrow \:2\le \frac{\left(a+b\right)^2}{4}\:\rightarrow \:\frac{1}{2}\le \frac{\left(a+b\right)^2}{16}\:\rightarrow \:\frac{3}{2\left(a+b\right)^2}\le \:\frac{3}{16}$

Answer 1

Why you found the maximal value of $a, b$? The question is asking $\frac3{2(a+b)^2}$!

Using AM-GM is, yes, the best approach. Here is my answer.

Since using AM-GM, $8=4ab\le(a+b)^2$, so $\frac1{(a+b)^2}\le\frac18$.

So the maxima of $\frac3{2(a+b)^2}$ is $\frac32\cdot\frac18=\frac3{16}$ when $a=b=\pm\sqrt2$.

Answer 2

Substitute the given restriction into the proposed expression.

Then your problem reduces to study a single-variable real-valued function: \begin{align*} \frac{3}{2(a + b)^{2}} & = \frac{3}{2\left(a + \dfrac{2}{a}\right)^{2}}\\\\ & = \frac{3a^{2}}{2(a^{2} + 2)^{2}}\\\\ & = \frac{3}{2}\left(\frac{a}{a^{2} + 2}\right)^{2} \end{align*}

Since the quadratic function is increasing, it suffices to study when the argument attains its max.

More precisely, according to the AM-GM inequality, we can conclude that \begin{align*} a^{2} + 2 \geq 2\sqrt{2a^{2}} = 2\sqrt{2}|a| \Longleftrightarrow \left|\frac{a}{a^{2} + 2}\right| \leq \frac{1}{2\sqrt{2}} \end{align*}

Hence the maximum value attained is given by: \begin{align*} \frac{3}{2(a + b)^{2}} \leq \frac{3}{2}\left(\frac{1}{2\sqrt{2}}\right)^{2} = \frac{3}{16} \end{align*}

In order to determine $a$ which corresponds to the maximum, you can solve the equation: \begin{align*} a^{2} - 2\sqrt{2}|a| + 2 = 0 \end{align*}

Once you know $a$, you also know $b$ and you are done.

Hopefully this helps !