Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable
Mark the correct alternative of the following. =1+2 \sin x+3 \cos ^{2} x, 0 \leq
Mark the correct alternative of the following. \( f(x)=1+2 \sin x+3 \cos ^{2} x, 0 \leq x \leq \) \( \frac{2 \pi}{3} \) is?
- A. Minimum at \( x=\pi / 2 \)
- B. Maximum at \( x=\sin ^{-1}(1 / \sqrt{3}) \)
- C. Minimum at \( x=\pi / 6 \)
- D. Maximum at \( \sin ^{-1}(1 / 6) \)
Step-by-step solution
Rewrite f(x)=1+2sinx+3cos²x=4+2sinx-3sin²x. Let t=sinx, then f(t)=4+2t-3t², a downward parabola. Critical points: f'(x)=2cosx-6cosx sinx=2cosx(1-3sinx)=0 → cosx=0 (x=π/2) or sinx=1/3. Second derivative test: at x=π/2, f''>0 → local minimum; at sinx=1/3, f''<0 → local maximum. Endpoints: f(0)=4, f(2π/3)≈3.482, f(π/2)=3, so global minimum at x=π/2.
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