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

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)   (Show Source): You can put this solution on YOUR website!

The left hand side is the function

Plug in x = 0

The result is negative.

Plug in x = 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:

Those two identities are based on
where n > 0

Answer by ikleyn(52879)   (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 ? "

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