1.

Letf'(sin x)lt0 and f''(sin x) gt0 forall x in (0,(pi)/(2)) and g(x) =f(sinx)+f(cosx) which of the following is true?

Answer»

g(x) is DECREASING in `((pi)/(4),(pi)/(2))`
g(x) increasing in `(0,(pi)/(4))`
g(x) is nonotonically increasing in `(0,(pi)/(2))`
none of these

Solution :`g(x)=f(SINX)COSX-f(cosx)sinx`
or `g(x)=f(sinx)sinx+COS^(2)xf(sinx)`
`f(cos x)sin^(2)x-f(cosxgt0forall x in (0,pi//2)`
[as it is given `f(sinx)=f(cos x (pi//2-x))lt0`
Thus g(x) is increasing in `(0,pi//2)`. Also` g(pi//4)=0`
or `g(x)gt0forallx in ((pi)/(4),(pi)/(2))`
and `g(x) ltforall x in (0,pi//4)`
Thus g(X) is decreasing in `(0,pi//4)`


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