Let the function \(f\left( \theta \right) = \left| {\begin{array}{*{20}{c}} {\sin \theta }&{\cos \theta }&{\tan \theta }\\ {\sin \left( {\frac{\pi }{6}} \right)}&{\cos \left( {\frac{\pi }{6}} \right)}&{\tan \left( {\frac{\pi }{6}} \right)}\\ {\sin \left( {\frac{\pi }{3}} \right)}&{\cos \left( {\frac{\pi }{3}} \right)}&{\tan \left( {\frac{\pi }{3}} \right)} \end{array}} \right|\)

Where \(\theta \in \left[ {\frac{\pi }{6},\frac{\pi }{3}} \right]\) and f’(θ) denote the derivative of f with respect to θ. Which of the following statements is/are TRUE?

(I) There exists \(\theta \in \left( {\frac{\pi }{6},\frac{\pi }{3}} \right)\) such that f’(θ) = 0

(II) There exists \(\theta \in \left( {\frac{\pi }{6},\frac{\pi }{3}} \right)\) such that f’(θ) ≠ 0

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  1. I only
  2. II only
  3. Both I and II
  4. Neither I nor II

Answer (Detailed Solution Below)

Option 3 : Both I and II
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Detailed Solution

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Rolle's theorem.

To apply Rolle's theorem following 3 conditions should be satisfied:

f(x) should be continuous in interval [a, b],

f(x) should be differentiable in interval (a, b), and

f(a)=f(b)

If these 3 conditions are satisfied simultaneously then, there exists at least one ′x′ such that f′(x)=0

Explanation:

Statement I: There exists \(\theta \in \left( {\frac{\pi }{6},\frac{\pi }{3}} \right)\) such that f’(θ) = 0

For the given question, it satisfies all the three conditions, therefore there exists at least one θ that gives f′(θ)=0

Therefore, statement II is true

Statement II: There exists \(\theta \in \left( {\frac{\pi }{6},\frac{\pi }{3}} \right)\) such that f’(θ) ≠ 0  

The given function is also not a constant function therefore for some θ, f′(θ) ≠ 0

Therefore, statement II is true
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