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f(X)= x÷sinx in (0,p÷2) check the monotonicity?

f(X)= x÷sinx in (0,p÷2) check the monotonicity?

Grade:Upto college level

1 Answers

bharat bajaj IIT Delhi
askIITians Faculty 122 Points
7 years ago
f(x)= x/sinx
df/dx = (sinx - xcosx)/ (sinx)^2

A 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)^2

tan 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.

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