AP EAMCET · Maths · Application of Derivatives
If \(\alpha\) and \(\beta\) are the least and the greatest values of \(f(x)=\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2\) for all \(x \in R\) respectively, then \(8(\alpha+\beta)=\)
- A \(\pi^2\)
- B \(11 \pi^2\)
- C \(9 \pi^2\)
- D \(25 \pi^2\)
Answer & Solution
Correct Answer
(B) \(11 \pi^2\)
Step-by-step Solution
Detailed explanation
\[ f(x)=\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2 \] Let \[ \sin ^{-1} x=a \text { and } \cos ^{-1} x=b \] Then, \[ \begin{aligned} f(x) & =a^2+b^2 \\ & =(a+b)^2-2 a b \end{aligned} \] Put the value of \(a\) and \(b\)…
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