Find the Monotonocity

Calculus Level 1

If f ( x ) = x 2 1 − cos ⁡ x f(x)=\dfrac{x^2}{1-\cos x} , where 0 < x < 1 0<x<1 . Which of the following describes the function of f ( x ) f(x) ?

It is a decreasing function None of these choices (i.e. sometimes increasing, sometimes decreasing) It is an increasing function

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Relevant wiki: Recognizing Trigonometric Graphs

1 − c o s x = 2 s i n 2 ( x 2 ) 1-cosx=2sin^2(\frac{x}{2}) therefore function is 1/2 ( x s i n ( x / 2 ) ) 2 {(\frac{x}{sin(x/2)})}^2

as x s i n ( x / 2 ) \frac{x}{sin(x/2)} is increasing function (because d / d x d/dx of x x is 1 and d / d x d/dx of s i n ( x / 2 ) sin(x/2) is c o s ( x / 2 ) cos(x/2) ) therefore our required function is also increasing in ( 0 , 1 ) (0,1)

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That's not the proper way!

Deepak Kumar - 4 years, 10 months ago

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why it isn't the proper way

A Former Brilliant Member - 4 years, 10 months ago

rate of increase of sin(x/2) is cos(x/2) and rate of increase of x is 1. As 1 > o r = c o s ( x / 2 ) 1>or=cos(x/2) therefore, our function is increasing.

A Former Brilliant Member - 4 years, 10 months ago

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