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

Options

A. `1/root3e`
B. `1/e^(6)`
C. `e^(3)`
D. `1/e^(3)`

Answer

4

Explanation

`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