Prove that the function f given byf(x) = log sin x" is increasing on ")0,pi/2) and decreasing on (pi/2,pi).

Prove that the function f given byf(x) = log sin x" is increasing on "

1.	Prove that the function f given byf(x) = log sin x" is increasing on ")0,pi/2) and decreasing on (pi/2,pi).
Answer» Solution :f(X) = log (sin x) ` rArrf'(x) = (COS x)/(sin x) = cos x` (a)f(x) is increasing. ` rArrf,(x) GT 0` ` rArrcot x gt 0` `RARRX in (0,pi/2)` `:.F(x)" is increasing in "(0,pi/2)`. (b)f(x) is decreasing. `rArrf'(x) lt 0` ` rArrcotx lt 0` ` rArrx in (pi/2, pi)` ` :. f(x)" is decreasing in "(pi/2, pi)`.

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Prove that the function f given byf(x) = log sin x" is increasing on ")0,pi/2) and decreasing on (pi/2,pi).

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