SATHEE: Application of Derivatives 4 Question 54

Application of Derivatives 4 Question 54

####57. Let $f (x) = \sin^{3} x + λ \sin^{2} x, - \frac{π}{2} < x < \frac{π}{2} \cdot$ Find the

$(1986, 5 M)$ intervals in which $λ$ 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 $y = f (x) = \sin^{3} x + λ \sin^{2} x, - \frac{π}{2} < x < \frac{π}{2}$

Let $\sin x = t$

$\begin{aligned} ∴ y = t^{3} + λ t^{2}, - 1 < t < 1 \\ \Rightarrow \frac{d y}{d t} = 3 t^{2} + 2 t λ = t (3 t + 2 λ) \end{aligned}$

For exactly one minima and exactly one maxima $d y / d t$ must have two distinct roots $\in (- 1, 1)$ .

$\begin{array}{llrl} \Rightarrow & t = 0 & and t & t = - \frac{2 λ}{3} \in (- 1, 1) \\ \Rightarrow & - 1 < - \frac{2 λ}{3} < 1 \\ \Rightarrow & - \frac{3}{2} < λ < \frac{3}{2} \\ \Rightarrow & λ \in - \frac{3}{2}, \frac{3}{2} \end{array}$

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Application of Derivatives 4 Question 54

Answer:

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