SOLUTION: By using the Intermediate Value Theorem, show that the equation: nth root of x = 1-x has at least one solution in the open interval (0,1) for any positive integer n>=3. >= mean

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Question 1200073: By using the Intermediate Value Theorem, show that the equation: nth root of x = 1-x has at least one solution in the open interval (0,1) for any positive integer n>=3.
>= means greater than or equal to
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

root%28n%2Cx%29+=+1-x

root%28n%2Cx%29-1%2Bx+=+0

f%28x%29+=+0

The left hand side is the function f%28x%29+=+root%28n%2Cx%29-1%2Bx

Plug in x = 0
f%28x%29+=+root%28n%2Cx%29-1%2Bx

f%280%29+=+root%28n%2C0%29-1%2B0

f%280%29+=+0-1%2B0

f%280%29+=+-1
The result is negative.

Plug in x = 1
f%28x%29+=+root%28n%2Cx%29-1%2Bx

f%281%29+=+root%28n%2C1%29-1%2B1

f%281%29+=+1-1%2B1

f%281%29+=+1
The result is positive.

f(x) changes from negative to positive when going through this interval 0 < x < 1.
Somewhere in the middle we must have f(x) = 0 occur at least once, which leads to f(x) having at least one root in this interval.
This is because f(x) is a continuous curve.

Identities used:
root%28n%2C0%29+=+0
root%28n%2C1%29+=+1
Those two identities are based on
0%5En+=+0 where n > 0
1%5En+=+1

Answer by ikleyn(52876) About Me  (Show Source):
You can put this solution on YOUR website!
.

Similar statement is valid even for  n = 2,  too,

so you can replace the inequality  n >= 3  in your post by more strong inequality  n >= 2.


Otherwise,  a knowledgeable person may ask   " why  n= 2  is not included ? "