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Math Help - Fixed Points- Proof

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    Fixed Points- Proof

    A number a is called a fixed point of a function f if f(a)=a. Assuming that f is differentiable and that for every x one has f'(x)does not equal1 prove that f has at most one fixed point.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by amm345 View Post
    A number a is called a fixed point of a function f if f(a)=a. Assuming that f is differentiable and that for every x one has f'(x)does not equal1 prove that f has at most one fixed point.
    What do you think?
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    I think I've got it, can you check this for me?

    Suppose f has two fixed point, x1 <x2. Then by MVT we have that
    y (x1,x2),f(x2) f(x1)= f0(y). x2 x1
    Then f0(y)=1 which is a contradiction.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by amm345 View Post
    I think I've got it, can you check this for me?

    Suppose f has two fixed point, x1 <x2. Then by MVT we have that
    y (x1,x2),f(x2) f(x1)= f0(y). x2 x1
    Then f0(y)=1 which is a contradiction.
    This is hard to decipher what you mean. Basically all you need to note is that if x_1\ne x_2 were both fixed points then the MVT guarantees a number in between the two where the derivative is one.
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