First slide

Maxima and minima

Question

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

Moderate

A

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

$(- \frac{1}{2}, \frac{1}{2}) - \{0\}$
C

$(- \frac{3}{2}, \frac{3}{2})$
D

$(- \frac{1}{2}, \frac{1}{2})$

Solution

$\begin{array}{l} f (x) = (1 - {Cos}^{2} x) (λ + \sin x) = {Sin}^{2} x (λ + \sin x) \\ f^{1} (x) = \sin^{2} x Cos x + (λ + \sin x) 2 \sin x \cos x \\ = \sin x \cos x (2 λ + 3 \sin x) \end{array}$
For maximum or minimum,
$\begin{array}{l} f^{1} (x) = 0 \Rightarrow \sin x = 0 or \sin x = \frac{- 2 λ}{3} \\ As there is exactly one max. and exactly one min, f^{'} (x) = 0 has exactly 2 roots \\ - 1 < \frac{2 λ}{3} < 1 and \frac{- 2 λ}{3} \neq 0 \Rightarrow λ \in (\frac{- 3}{2}, \frac{3}{2}) - {0} \end{array}$

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