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6. Application of derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths6. Application of derivatives1 Mark
MCQ
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
    $(-3,1)$
  • ✓
    $\left[-\frac{4}{3}, 2\right]$
  • C
    $[1, \infty)$
  • D
    $(-\infty,-1]$
✓

Answer

Correct option: B.
$\left[-\frac{4}{3}, 2\right]$
b
$f(x)=(4 a-3)\left(x+\log _{e} 5\right)+(a-7) \sin x$

$f'(x)=(4 a-3)(1)+(a-7) \cos x=0$

$\Rightarrow \quad \cos x=\frac{3-4 a}{a-7}$

$-1 \leq \frac{3-4 a}{a-7}<1$

$\frac{3-4 a}{a-7}+1 \geq 0$

$\frac{3-4 a+a-7}{a-7} \geq 0$

$\frac{-3 a-4}{a-7} \geq 0$

$\frac{3 a+4}{a-7} \leq 0$

$\frac{3-4 a}{a-7}<1$

$\frac{3-4 a}{a-7}-1<0$

$\frac{3-4 a-a+7}{a-7}<0$

$\frac{-5 a+10}{a-7}<0$

$\frac{5 a-10}{a-7}>0$

$\frac{5(a-2)}{a-7}>0$

$\alpha \in\left[-\frac{4}{3}, 2\right)$

Check end point $\left[-\frac{4}{3}, 2\right)$

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