Home
Class 12
MATHS
Let f(x)=sin^3x+lambda sin^2x , -pi/2 < ...

Let `f(x)=sin^3x+lambda sin^2x , -pi/2 < x < pi/2` Find the intervals in which `lambda` should lie in order that `f(x)` has exactly one minimum and exactly one maximum.

Text Solution

Verified by Experts

We have
`f(x)=2(x-1)(x-2)3+3(-1)2(x-2)^(2)2`
`=(x-1)(x-2)^(2)(5x-7)`
Sign schem of f(x) is as shown in the following figure

Clearly f(x) has point of maxima at x=1 and point of minima t `x=7//5`
Also sign of derivative is not changing at x=2 so it is not a point of extrema.
Promotional Banner

Topper's Solved these Questions

  • MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS

    CENGAGE ENGLISH|Exercise Solved Examples|20 Videos

  • MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS

    CENGAGE ENGLISH|Exercise Concept Application Exercise 6.1|10 Videos

  • METHODS OF DIFFERENTIATION

    CENGAGE ENGLISH|Exercise Single Correct Answer Type|46 Videos

  • MONOTONOCITY AND NAXINA-MINIMA OF FUNCTIONS

    CENGAGE ENGLISH|Exercise Comprehension Type|6 Videos

Similar Questions

Explore conceptually related problems

Let f(x) = sin^3x+lambdasin^2x , pi/2 < x < pi/2dot Find the intervals in which lambda should lie in order that f(x) has exactly one minimum and exactly one maximum.

Let f(x)=2sin^(3)x+lamdasin^(2)x,-(pi)/2ltxlt(pi)/2 . If f(x) has exactly one minimum and one maximum, then lamda cannot be equal to

Let f(x)=-sin^3x+3sin^2x+5 on [0,pi/2] . Find the local maximum and local minimum of f(x)dot

Separate the intervals of monotonocity of the function: f(x)=-sin^3x+3sin^2x+5,x in [-pi/2,pi/2]dot

Let f(x)={(1-sin^3x)/(3cos^2x) if x pi/2 find a and b. .

Let f(x)=|x|+|sin x|, x in (-pi//2,pi//2). Then, f is

Find the maximum and the minimum values of f(x)=sin3x+4,\ \ x in (-pi//2,\ pi//2) , if any.

The period of the function f(x)=c^((sin^2x) +sin^2 (x+pi/3)+cosxcos(x+pi/3)) is (where c is constant)

Let f(x) = |(2cos^2x, sin2x, -sinx), (sin2x, 2 sin^2x, cosx), (sinx, -cosx,0)| , then the value of int_0^(pi//2){f(x) + f'(x)} dx is

Let the function f(x) be defined as below f(x)=sin^(-1) lambda+x^2 when 0 =1 . f(x) can have a minimum at x=1 then value of lambda is