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 :
Correct Option: , 4
$f(x)=\left(1-\cos ^{2} x\right)(\lambda+\sin x)$
$x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$
$f(x)=\lambda \sin ^{2} x+\sin ^{3} x$
$f^{\prime}(x)=2 \lambda \sin x \cos x+3 \sin ^{2} x \cos x$
$f^{\prime}(x)=\sin x \cos x(2 \lambda+3 \sin x)$
$\sin x=0, \frac{-2 \lambda}{3},(\lambda \neq 0)$
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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