Application of Derivatives 4 Question 54
57. Let $f(x)=\sin ^{3} x+\lambda \sin ^{2} x,-\frac{\pi}{2}<x<\frac{\pi}{2} \cdot$ Find the
$(1986,5 M)$ intervals in which $\lambda$ should lie in the order that $f(x)$ has exactly one minimum and exactly one maximum.
$(1985,5 M)$
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Answer:
Correct Answer: 57. (9)
Solution:
- Let $y=f(x)=\sin ^{3} x+\lambda \sin ^{2} x,-\frac{\pi}{2}<x<\frac{\pi}{2}$
Let $\quad \sin x=t$
$$ \begin{aligned} & \therefore \quad y=t^{3}+\lambda t^{2},-1<t<1 \\ & \Rightarrow \quad \frac{d y}{d t}=3 t^{2}+2 t \lambda=t(3 t+2 \lambda) \end{aligned} $$
For exactly one minima and exactly one maxima $d y / d t$ must have two distinct roots $\in(-1,1)$.
$$ \begin{array}{llrl} \Rightarrow & t=0 & \text { and } t & t=-\frac{2 \lambda}{3} \in(-1,1) \\ \Rightarrow & -1<-\frac{2 \lambda}{3}<1 \\ \Rightarrow & -\frac{3}{2}<\lambda<\frac{3}{2} \\ \Rightarrow & \lambda \in-\frac{3}{2}, \frac{3}{2} \end{array} $$
