Inverse Trigonometry
Optimization of expression involving cot⁻¹
MMTS_Full_Test_19
Grade 12
Question:
The maximum value of the function $f(x) = \dfrac{4\cot^{-1}x}{\pi} - \dfrac{\pi}{4\cot^{-1}(-x)}$ occurs at $x$ equal to
(A) $-1$
(B) $0$
(C) $1$
(D) None of these
Step-by-Step Solution
Key Concept: Let $\theta = \cot^{-1}(-x) = \pi - \cot^{-1}(x)$. Rewrite $f$ in terms of $\theta$ and apply AM-GM.
Maximum $f(-1)=2$ by AM-GM, equality at $\cot^{-1}(-x)=\frac{\pi}{4} \Rightarrow x=-1$.
Correct Answer: (A) $-1$