MHB What Is the Maximum Value of \(a^2+ab+2b^2\) Given \(a^2-ab+2b^2=8\)?

[anemone]

$a$ and $b$ are positive real numbers such that $a^2-ab+2b^2=8$.

Find the maximum value of $a^2+ab+2b^2$.

 

[poissonspot]

Here:

Spoiler

$\frac{4}{7}(4+\sqrt2)^2$

 

[anemone]

Here:

Spoiler

$\frac{4}{7}(4+\sqrt2)^2$

Your answer is correct, conscipost! (Yes) Well done!

But I'd appreciate it if you show your solution (but not merely the final answer) so we know what approach you used, sounds good to you?