Question

How do we find whether the function `f(x)=cosx-x^2` has a root between `x=pi/4` and `x=pi/3` or not?

Answer 1

Please follow as given below.

Explanation:

When we have two points connected by a continuous curve, one point below the line and the other above the line, then according to Intermediate Value Theorem, there will be at least one point where the curve crosses the line.

As at `x=pi/4`,

`f(x)=cos(pi/4)-(pi/4)^2=1/sqrt2-pi^2/16=0.7071-0.61685=0.09025`

As at `x=pi/3`,

`f(x)=cos(pi/3)-(pi/3)^2=1/2-pi^2/9=0.5-1.09662=-0.59662`

As between `x=pi/4` and `x=pi/3`, `f(x)` has moved from negative to positive, `f(x)` has at least one value of `x`, where `f(x)=0`

and hence,

the function `f(x)=cosx-x^2` has a root between `x=pi/4` and `x=pi/3`.