if-the-range-of-f-x-y-sec-1-x-1-x-sec-1-y-1-y-xy-lt-0-is-a-b-and-a-b-equals-pi-10-then-is-eqal-to-

Question Number 68991 by pranay02 last updated on 17/Sep/19

i f t h e r a n g e o f f (x, y) = {s e c}^{- 1} (x + \frac{1}{x}) + {s e c}^{- 1} (y + \frac{1}{y}), x y < 0 i s (a, b) a n d (a + b) e q u a l s \frac{λ π}{10}, t h e n λ i s e q a l t o

Answered by MJS last updated on 18/Sep/19

g (t) = \sec^{- 1} (t + \frac{1}{t}) = \cos^{- 1} (\frac{t}{t^{2} + 1})

\frac{π}{3} ⩽ g (t) ⩽ \frac{2 π}{3}

the minimum is at (\begin{matrix} 1 \\ \frac{π}{3} \end{matrix})

the maximum is at (\begin{matrix} - 1 \\ \frac{2 π}{3} \end{matrix})

t < 0 \Rightarrow \frac{π}{2} < g (t) ⩽ \frac{3 π}{2}

t > 0 \Rightarrow \frac{π}{3} ⩽ g (t) < \frac{π}{2}

\Rightarrow \frac{5 π}{6} < f (x, y) < 2 π for x y < 0

\frac{5 π}{6} + 2 π = \frac{17 π}{6} = \frac{λ π}{10} \Rightarrow λ = \frac{85}{3}

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