JEE PYQ: Application Of Derivatives Question 1
Question 1 - 2021 (16 Mar Shift 1)
The range of $a \in \mathbb{R}$ for which the function
$$f(x) = (4a - 3)(x + \log_e 5) + 2(a - 7)\cot\left(\frac{x}{2}\right)\sin^2\left(\frac{x}{2}\right)$$
$x + 2n\pi$, $n \in \mathbb{N}$, has critical points, is
(1) $(-3, 1)$
(2) $\left[-\frac{4}{3}, 2\right)$
(3) $[1, \infty)$
(4) $(-\infty, -1]$
Show Answer
Answer: (2)
Solution
$f’(x) = (4a-3)(1) + (a-7)\sin x = 0 \Rightarrow \cos x = \frac{3-4a}{a-7}$, requiring $-1 \le \frac{3-4a}{a-7} \le 1$. Solving the two inequalities gives $\frac{-3a-4}{a-7} \ge 0$ and $\frac{5(a-2)}{a-7} > 0$, yielding $a \in \left[-\frac{4}{3}, 2\right)$.
BETA
