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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