Question

The maximum value of $\left(2\right(a-x\left)\right(x+\sqrt{x^2+b^2})\forall x,a,b\in R)$ is :

• A. $a^2-b^2$
• B. $a^2+b^2$
• C. $a-b$
• D. $a+b$

Answer 1

The correct option is B

$a^2+b^2$

Given: $2(a-x)(x+\sqrt{x^2+b^2})$
put $t=x+\sqrt{x^2+b^2}$
$\Rightarrow \sqrt{x^2+b^2}=t-x$
$\Rightarrow x^2+b^2=t^2+x^2-2tx$
$\Rightarrow 2tx=t^2-b^2$
$\Rightarrow x=\frac{t^2-b^2}{2t}$

$\therefore y=2(a-x)t$
$\Rightarrow y=2at-2xt$
$\Rightarrow y=2at-(t^2-b^2)$
$\Rightarrow y=2at-(t^2-b^2)+(a^2-a^2)$
$\Rightarrow y=(a^2+b^2)-(a^2+t^2-2at)$
$\Rightarrow y=(a^2+b^2)-(t-a{)}^2$
$\Rightarrow y\le$ $a^2+b^2$