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15/01/2022 3:52 pm
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Let M and m respectively be the maximum and minimum values of the function f(x) = tan-1 (sinx + cosx) in [0, π/2], Then the value of tan(M - m) is equal to
(1) 2 + √3
(2) 2 - √3
(3) 3 + 2√2
(4) 3 - 2√2
1 Answer
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15/01/2022 4:03 pm
Correct answer: (4) 3 - 2√2
Explanation:
Let g(x) = sin x + cos x = √2 sin (x + π/4)
g(x) ∈ [1, √2] for x ∈ [0, π/2]
f(x) = tan-1 (sinx + cosx) ∈ [π/4, tan-1√2]
tan(tan-1√2 - π/4) = \(\frac{\sqrt2 - 1}{1 + \sqrt2}\) x \(\frac{\sqrt2 - 1}{\sqrt2 - 1}\)
= 3 - 2√2