COMEDK · Maths · 27. Application of Derivatives
\(\text { The function } y=\tan x-x \text { is }\)
- A \(\text { decreasing in }\left(0, \dfrac{\pi}{4}\right) \text { and increasing in }\left(\dfrac{\pi}{4}, \dfrac{\pi}{2}\right)\)
- B \(\text { a decreasing function in }\left(0, \dfrac{\pi}{2}\right)\)
- C \(\text { an increasing function in }\left(0, \dfrac{\pi}{2}\right)\)
- D \(\text { increasing in }\left(0, \dfrac{\pi}{4}\right) \text { and decreasing in }\left(\dfrac{\pi}{4}, \dfrac{\pi}{2}\right)\)
Answer & Solution
Correct Answer
(C) \(\text { an increasing function in }\left(0, \dfrac{\pi}{2}\right)\)
Step-by-step Solution
Detailed explanation
Let \(f(x) = \tan x - x\). To determine the intervals of increase or decrease, find the first derivative of \(f(x)\) with respect to \(x\). \(f'(x) = \dfrac{d}{dx}(\tan x - x) = \sec^2 x - 1\). Using the trigonometric identity \(\sec^2 x - 1 = \tan^2 x\), we have…
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