GATE | GATE-CS-2014-(Set-1) | Question 16

GATECS2014Q16
(A) I only
(B) II only
(C) Both I and II
(D) Neither I nor II


Answer: (C)

Explanation: In the given question,\theta ranges from \pi /6 to \pi /3.
When \theta=\pi /6, the value of the function is 0, since Rows 1 and 2 are the same.
When \theta=\pi /3, the value of the function is 0, since Rows 1 and 3 are the same.
Therefore, f(\pi /6) = f(\pi /3) =0
The given function is continuous and differentiable in the given range since it is a trigonometric function.
Then According to the Mean Value Theorem,
There exists a value c of such that f'(c)=0. So the first statement is true.
Also there may also exist a value c, such that f'(c)\neq 0. This is because the function is not a constant function, i.e. all values in the range are not equal to c.
So the second statement is also true.
Therefore the correct option is C.

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