The set of all real values of for which the function f(x)= (1- ^ … - Vidyadip

MCQ

The set of all real values of $\lambda$ for which the function $f(x)=\left(1-\cos ^{2} x\right) \cdot(\lambda+\sin x)$ $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right),$ has exactly one maxima and exactly one minima, is

A
$\left(-\frac{1}{2}, \frac{1}{2}\right)-\{0\}$
B
$\left(-\frac{1}{2}, \frac{1}{2}\right)$
C
$\left(-\frac{3}{2}, \frac{3}{2}\right)$
✓
$\left(-\frac{3}{2}, \frac{3}{2}\right)-\{0\}$

✓

Answer

Correct option: D.

$\left(-\frac{3}{2}, \frac{3}{2}\right)-\{0\}$

d
$f(x)=\left(1-\cos ^{2} x\right)(\lambda+\sin x)$

$x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$

$f(x)=\lambda \sin ^{2} x+\sin ^{3} x$

$f^{\prime}(x)=2 \lambda \sin x \cos x+3 \sin ^{2} x \cos x$

$f^{\prime}(x)=\sin x \cos x(2 \lambda+3 \sin x)$

$\sin x=0, \frac{-2 \lambda}{3}, \quad(\lambda \neq 0)$

for exactly one maxima \& minima

$\frac{-2 \lambda}{3} \in(-1,1) \Rightarrow \lambda \in\left(\frac{-3}{2}, \frac{3}{2}\right)$

$\lambda \in\left(-\frac{3}{2}, \frac{3}{2}\right)-\{0\}$

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