Let the function f (\theta ) = \left|\begin{array}{c}\sin \\\⁡\theta & \cos ⁡\theta & \tan ⁡\theta \ \sin \\\⁡ (\\\{\pi\\\\end\frac{array}{6}) , \cos ⁡ (\frac{\pi }{6}) , \tan ⁡ (\frac{\pi }{6}) ; \sin ⁡ (\frac{\pi }{3}) , \cos ⁡ (\frac{\pi }{3}) , \tan ⁡ (\frac{\pi }{3}) ; |} Where \theta \in [\frac{\pi }{6},\frac{\pi }{3}] and f’(θ) denote the derivative of f with respect to θ. Which of the following statements is/are TRUE? (I) There exists \theta \in (\frac{\pi }{6},\frac{\pi }{3}) such that f’(θ) = 0 (II) There exists \theta \in (\frac{\pi }{6},\frac{\pi }{3}) such that f’(θ) ≠ 0

Correct Answer is : Both I and II

GATE Computer Science (CS) 2014 Shift 1 Solved Paper

Question : 16 of 65

Marks: +1, -0

Let the function

f (θ ) = {|\table sin \;⁡θ , \cos ⁡θ , \tan ⁡θ ; sin \;⁡ (\;{π\;}/{6}) , \cos ⁡ (\;{π\;}/{6}) , \tan ⁡ (\;{π\;}/{6}) ; sin \;⁡ (\;{π\;}/{3}) , \cos ⁡ (\;{π\;}/{3}) , \tan ⁡ (\;{π\;}/{3}) ; |}

Where

θ ∈ [\;{π\;}/{6},\;{π\;}/{3}]

and f’(θ) denote the derivative of f with respect to θ. Which of the following statements is/are TRUE?
(I) There exists

θ ∈ (\;{π\;}/{6},\;{π\;}/{3})

such that f’(θ) = 0
(II) There exists

θ ∈ (\;{π\;}/{6},\;{π\;}/{3})

such that f’(θ) ≠ 0

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