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  4. Let f(x)= sin -1 x √1-x-√x(1-x2) ∀ 0 ≤ x ≤ 1, then f(x) is

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

NTA AbhyasNTA Abhyas 2022

Solution:

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