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Check the monotonicity of f (x) = x/sin x in (0, p/2).

Check the monotonicity of f (x) =
x/sin x
in (0, p/2).

Grade:12

2 Answers

Sher Mohammad IIT Delhi
askIITians Faculty 174 Points
7 years ago
f'(x)=sinx-xcosx/(sinx)^2

for small x, sinx=x
f'(x)=x(1-cos x)> greater than zero
for higher values of x, also f'(x)>0, hence the function f(x) is monotonically increasing.

Sher Mohammad
B.Tech, IIT Delhi




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.
Thanks & Regards
Bharat Bajaj
askIITians Faculty
IIT Delhi

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