Question

Let $f(x)={\sin }^{-1}\left\{x\sqrt{1-x}-\sqrt{x\left(1-x^2\right)}\right\},\forall 0\le x\le 1$, then $f(x)$ is

• A. negative
• B. positive
• C. non-negative
• D. non-positive

Answer 1

Answer: (D)

$f(x)={\sin }^{-1}\left(x\sqrt{1-(\sqrt{x}{)}^2}-\sqrt{x}\sqrt{1-x^2}\right),\forall 0\le x\le 1$
$\because {\sin }^{-1}(a)-{\sin }^{-1}(b)={\sin }^{-1}\left(a\sqrt{1-b^2}-b\sqrt{1-a^2}\right)$
$\Rightarrow f(x)={\sin }^{-1}x-{\sin }^{-1}\sqrt{x}\le 0$ (as $x\le \sqrt{x})$