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Math Help - differentiability and continuity problem

  1. #1
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    differentiability and continuity problem

    #1 Suppose that the differentiable function f(x) satisfies f(1)=2 and f'(x) \leq 1. How large can f(3) possibly be?(Done)
    If f(3)=4, then show that f(x)=x+1 for 1 \leq x \leq  3.

    How to approach this?

    #2 Assume f(x) is continuous on [a,b], and for any x\in[a,b], there exists y\in[a,b] such that |f(y)| \leq  \frac{1}{2}|f(x)|. Prove that there exists a point c \in [a,b] such that f(c)=0.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: differentiability and continuity problem

    Quote Originally Posted by maoro View Post
    #1 Suppose that the differentiable function f(x) satisfies f(1)=2 and f'(x) \leq 1. How large can f(3) possibly be?(Done)
    If f(3)=4, then show that f(x)=x+1 for 1 \leq x \leq  3.
    Using the Lagrange Theorem for f on [1,3] we have f(3)=2f'(\xi)+f(1) for some \xi\in(1,3) . If f(1)=2 , as f'(\xi)\leq 1 we conclude f(3)\leq 4 . If f(3)=4 then f'(\xi)=1 so ... (conclude)
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    Re: differentiability and continuity problem

    I have just learned about Mean Value Theorem, is it similar to Lagrange Theorem?
    btw, if f(3)=4, we can just show that f'(\xi)=1, but not f(x)=x+1
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    Re: differentiability and continuity problem

    Quote Originally Posted by maoro View Post
    #2 Assume f(x) is continuous on [a,b], and for any x\in[a,b], there exists y\in[a,b] such that |f(y)| \leq  \frac{1}{2}|f(x)|. Prove that there exists a point c \in [a,b] such that f(c)=0.
    Hint: g(x)=|h(x)| is continuous on [a,b] .
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    MHF Contributor FernandoRevilla's Avatar
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    Re: differentiability and continuity problem

    Quote Originally Posted by maoro View Post
    I have just learned about Mean Value Theorem, is it similar to Lagrange Theorem?
    Yes, it is another name.

    btw, if f(3)=4, we can just show that f'(\xi)=1, but not f(x)=x+1
    You can prove that f'(x)=1 for all x\in (1,3) so f(x)=x+C in (1,3) etc ...
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    Re: differentiability and continuity problem

    Quote Originally Posted by FernandoRevilla View Post
    Yes, it is another name.



    You can prove that f'(x)=1 for all x\in (1,3) so f(x)=x+C in (1,3) etc ...
    Is it because f is differentiable, then if f'(x)=1, we get f(x)=x+C ?
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    MHF Contributor FernandoRevilla's Avatar
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    Re: differentiability and continuity problem

    Quote Originally Posted by maoro View Post
    Is it because f is differentiable, then if f'(x)=1, we get f(x)=x+C ?
    Right.
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    Re: differentiability and continuity problem

    Quote Originally Posted by FernandoRevilla View Post
    Hint: g(x)=|h(x)| is continuous on [a,b] .
    What's the next step when |f(x)| is continuous on [a,b] ?
    Can i simply state that there exists c\in [a,b] such that

    |f(c)|\leq \frac{1}{2}|f(x_1)|\leq \frac{1}{2^2}|f(x_2)|\leq...\leq\frac{1}{2^n}|f(x_  n)|
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    MHF Contributor FernandoRevilla's Avatar
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    Re: differentiability and continuity problem

    Quote Originally Posted by maoro View Post
    |f(c)|\leq \frac{1}{2}|f(x_1)|\leq \frac{1}{2^2}|f(x_2)|\leq...\leq\frac{1}{2^n}|f(x_  n)|
    You have to prove that c exists, so don't start with it. Construct a sequence (y_n) in [a,b] such that 0\leq g(y_n)\leq  2^{-n}g(a) , so \lim_{n\to +\infty}g(y_n)=0 etc.
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    Re: differentiability and continuity problem

    I saw another problem which is similar to #1
    Suppose that f(x)=0 and f'(x) \leq 1. How large can f(4) possibly be? (Done)
    If f(4) =4, then show that f(x)=x for 0 \leq x \leq 4

    The question i want to ask is how can i prove f(x)=x if i don't know f is differentiable ?
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  11. #11
    MHF Contributor FernandoRevilla's Avatar
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    Re: differentiability and continuity problem

    Quote Originally Posted by maoro View Post
    The question i want to ask is how can i prove f(x)=x if i don't know f is differentiable ?
    You can't, there are infinite functions satisfying f(0)=0 and f(4)=4 .
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