Question

Let $f\left(x\right)={\int }_1^x\sqrt{2-t^2}dt.$Then real roots of the equation $x^2-f^{\prime }\left(x\right)=0$ are

• A. $\pm 1$
• B. $\pm \frac{1}{\sqrt{2}}$
• C. $\pm \frac{1}{2}$
• D. 0 and 1

Answer 1

The correct option is A $\pm 1$

$f\left(x\right)={\int }_1^x\sqrt{2-t^2}dt{f}^{\prime }\left(x\right)=\sqrt{2-x^2}$

Also $x^2-f^{\prime }\left(x\right)=0$
$\therefore x^2=\sqrt{2-x^2}\Rightarrow x^4=2-x^2\Rightarrow x^4+x^2-2=0\Rightarrow x=\pm 1$