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Let f(x)=sin^3x+lambdasin^2x ,-pi/2<x<pi...

Let `f(x)=sin^3x+lambdasin^2x ,-pi/2

Text Solution

Verified by Experts

The correct Answer is:
`lamda in (- (3)/(2), (3)/(2))`
Let ` y = f(x)= sin ^(3) x + lamda sin ^(2) x, - (pi)/(2) lt x lt (pi)/(2)`
Let ` " " sin x = t `
`therefore " " y = t ^(3) + lamda t ^(2), - 1 lt t lt 1 `
`rArr " " (d y)/(dt) = 3 t^(2) + 2 t lamda = t ( 3t + 2 lamda)`
For exactly one minima and exactly one maxima ` d y //dt ` must have two distinct roots ` in ( -1, 1 )`
` rArr " " t = 0 and t = - ( 23)/( 3) in ( -1, 1 )`
`rArr " " - 1 lt - ( 2lamda )/(3) lt 1 `
`rArr " " - ( 3)/(2) lt lamda lt (3)/(2)`
`rArr " " lamda in (- (3)/(2), (3)/(2))`
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