Let the function \(f(\theta) = \begin{vmatrix} \sin\theta & \cos\theta &…
2014
Let the function
\(f(\theta) = \begin{vmatrix} \sin\theta & \cos\theta & \tan\theta \\ \sin(\frac{\pi}{6}) & \cos(\frac{\pi}{6}) & \tan(\frac{\pi}{6}) & \\ \sin(\frac{\pi}{3}) & \cos(\frac{\pi}{3}) & \tan(\frac{\pi}{3}) \end{vmatrix}\)
where \(\theta \in \left[ \frac{\pi}{6},\frac{\pi}{3} \right]\) and \(f'(\theta)\) denote the derivative of ݂\(f\) with respect to \(\theta\). Which of the following statements is/are TRUE?
I. There exists \(\theta \in (\frac{\pi}{6},\frac{\pi}{3})\) such that \(f'(\theta) = 0\)
II. There exists \(\theta \in (\frac{\pi}{6},\frac{\pi}{3})\) such that \(f'(\theta)\neq 0\)
- A.
I only
- B.
II only
- C.
Both I and II
- D.
Neither I nor II
Attempted by 31 students.
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Correct answer: C
Key idea: Use linearity of the determinant in the first row and analyze the derivative on the interval.
Write the determinant as a linear combination of the first-row entries. With the fixed second and third rows, one gets
f(θ) = D1·sinθ - D2·cosθ - (1/2)·tanθ,
where
D1 = det([[cos(π/6), tan(π/6)],[cos(π/3), tan(π/3)]]) = (3√3 - 1)/(2√3) ≈ 1.211
D2 = det([[sin(π/6), tan(π/6)],[sin(π/3), tan(π/3)]]) = (√3 - 1)/2 ≈ 0.366
Differentiate:
f'(θ) = D1·cosθ + D2·sinθ - (1/2)·sec²θ,
Evaluate at θ = π/6: cos(π/6)=√3/2, sin(π/6)=1/2, sec²(π/6)=4/3, so
f'(π/6) ≈ 1.211·0.866 + 0.366·0.5 - 0.5·(4/3) ≈ 0.565 > 0.
Evaluate at θ = π/3: cos(π/3)=1/2, sin(π/3)=√3/2, sec²(π/3)=4, so
f'(π/3) ≈ 1.211·0.5 + 0.366·0.866 - 0.5·4 ≈ -1.078 < 0.
Because f' is continuous on [π/6, π/3] and changes sign between the endpoints, the Intermediate Value Theorem guarantees at least one θ in (π/6, π/3) with f'(θ)=0. This proves the existence required in the first statement.
To see that there also exists a θ with f'(θ) ≠ 0, evaluate at θ = π/4: cos(π/4)=sin(π/4)=√2/2 and sec²(π/4)=2, giving
f'(π/4) ≈ (D1 + D2)·0.7071 - 1 ≈ 0.115 ≠ 0.
Therefore both statements are true: there exists at least one point in the open interval where f'(θ)=0, and there also exists at least one point in the open interval where f'(θ)≠0.