Use the intermediate Value Theorem to show that x^(1/3)=1-x have at least a solution in [0,1]?

1 Answer
F. Javier B.
May 25, 2018

Let be the function f(x)=root(3)x-1+x

This function is continous in RR even in [0,1]

Now evaluate the function in extreme points

f(0)=0-1+0=-1<0
f(1)=1-1+1=1>0

The intermediate value theorem (based on Bolzano's theorem) establish that exist a number c in [0,1] such that f(c)=0

And this is what we want to proof. QED

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