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Check the monotonicity of f (x) =x/sin xin (0, p/2).
f'(x)=sinx-xcosx/(sinx)^2for small x, sinx=xf'(x)=x(1-cos x)> greater than zerofor higher values of x, also f'(x)>0, hence the function f(x) is monotonically increasing.Sher MohammadB.Tech, IIT Delhi
f(x)= x/sinxdf/dx = (sinx - xcosx)/ (sinx)^2A function is monotonic if it is either increasing or decreasing in the given range.Hence, check the sign of df/dx.df/dx = (tanx - x)cosx/(sinx)^2tan x - x> 0 for (0,pi/2) . You may check it by making the graph.cosx>0 for (0,pi2)Hence, df/dx>0 for(0,pi/2)Hence, this function is monotonic in the given range.Thanks & RegardsBharat BajajaskIITians FacultyIIT Delhi
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