Question

Use Rolle's theorems to show that there is a solution of the equation cotx=x in ]0,π/2[

Answer 1

Let:

$f(x)=cosx-xsinx$

This function is continuous on the interval $[0,\pi/2]$

$f(0)=1, f(\pi/2)=-\pi/2$

So, by Intermediate Value Theorem there is $c\isin[0,\pi/2]$ , such that $f(c)=0$

That is, $c$ is the solution of the given equation.