Question

How do we know that the equation \[x + \sin(x) + \sqrt{x} = \pi\] has a solution?

Answer 1

Let \[f(x)=x +\sin(x)+ \sqrt{x}- \pi\]
the domain is [0,inf)
the function is continous
if we can show that for some number a
f(a)=negative number
and for some number b
f(b)=positive number
then there must be a c an element of (a,b) such f(c)=0
so if we can show this then that means f has a solution since it would be it crosses the x-axis

Answer 2

f(1)=1+sin(1)+1^(1/2)-pi=-1.12
f(pi)=pi+sin(pi)+pi^(1/2)-pi=pi+pi^(1/2)-pi=pi^(1/2)
so there is a number c between 1 and pi such that
f(c)=0

Answer 3

Thank you.