JEE Mains · Maths · STD 12 - 6. Application of derivatives
The range of \(a \in R\) for which the function \( f(x)=(4 a-3)\left(x+\log _{e} 5\right)+2(a-7) \cot \left(\frac{x}{2}\right) \sin ^{2}\left(\frac{x}{2}\right)\) \(x \neq 2 n \pi, n \in N ,\) has critical points, is
- A \((-3,1)\)
- B \(\left[-\frac{4}{3}, 2\right]\)
- C \([1, \infty)\)
- D \((-\infty,-1]\)
Answer & Solution
Correct Answer
(B) \(\left[-\frac{4}{3}, 2\right]\)
Step-by-step Solution
Detailed explanation
\(f(x)=(4 a-3)\left(x+\log _{e} 5\right)+(a-7) \sin x\) \(f'(x)=(4 a-3)(1)+(a-7) \cos x=0\) \(\Rightarrow \quad \cos x=\frac{3-4 a}{a-7}\) \(-1 \leq \frac{3-4 a}{a-7}<1\) \(\frac{3-4 a}{a-7}+1 \geq 0\) \(\frac{3-4 a+a-7}{a-7} \geq 0\) \(\frac{-3 a-4}{a-7} \geq 0\)…
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