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Least value of the function f(x)=e^(sinx...

Least value of the function f(x)=`e^(sinx-2sin^(2)x` is

A

`1/root3e`

B

`1/e^(6)`

C

`e^(3)`

D

`1/e^(3)`

Text Solution

Verified by Experts

The correct Answer is:
4
`e^(f(x)` is increasing or decreasing according as f(x) increases or decreases
Let f(x)=sinx-`2sin^(2)x`
`rArrf'(x)=cosx-4sinxcosx=(1-4sinx)cosx`
`f(x)=0rArrsinx=1/4orsinx=pm1`
(corresponding to cos x=0)
Also, f"(x)=-sinx-4cos2x
=-sinx-4+8`sin^(2)x`
f"(x)`gt`0 if sinx=`pm1` and f"(x)`lt`0 if sin x =`1/4`
Minima occurs when sin x=`pm1`
when sin x =1 then f(x)=-1
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Knowledge Check

  • The gratest value of the function f(x)=(sinx)/(sin(x+pi/4)) on thhe interval [0,pi//2] is

    A
    `1//sqrt2`
    B
    `sqrt2`
    C
    1
    D
    `-sqrt2`

  • The difference between the greatest and least values of the function f(x) = sin 2 x - x on [ -pi//2, pi//2] is

    A
    `pi`
    B
    `sqrt3/2 + pi/3`
    C
    `(sqrt3 + sqrt2)/2 + pi/6`
    D
    `1/2 (sqrt3 + sqr2)`

  • The difference between the greatest and least values of the function f(x) = sin 2x - x on [-pi//2 , pi//2] is

    A
    `(sqrt3 + sqrt2 )/(2)`
    B
    `(sqrt3 + sqrt2)/(2) + (pi)/(6)`
    C
    `(pi)/(2)`
    D
    `pi`

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