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Thread: differentiability and continuity problem

  1. #1
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    differentiability and continuity problem

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

    How to approach this?

    #2 Assume $\displaystyle f(x)$ is continuous on $\displaystyle [a,b]$, and for any $\displaystyle x\in[a,b]$, there exists y\in[a,b] such that $\displaystyle |f(y)| \leq \frac{1}{2}|f(x)|$. Prove that there exists a point $\displaystyle c \in [a,b]$ such that $\displaystyle f(c)=0.$
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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
    #1 Suppose that the differentiable function $\displaystyle f(x)$ satisfies $\displaystyle f(1)=2 $ and $\displaystyle f'(x) \leq 1$. How large can $\displaystyle f(3)$ possibly be?(Done)
    If $\displaystyle f(3)=4$, then show that $\displaystyle f(x)=x+1$ for $\displaystyle 1 \leq x \leq 3$.
    Using the Lagrange Theorem for $\displaystyle f$ on $\displaystyle [1,3]$ we have $\displaystyle f(3)=2f'(\xi)+f(1)$ for some $\displaystyle \xi\in(1,3)$ . If $\displaystyle f(1)=2$ , as $\displaystyle f'(\xi)\leq 1$ we conclude $\displaystyle f(3)\leq 4$ . If $\displaystyle f(3)=4$ then $\displaystyle f'(\xi)=1$ 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 $\displaystyle f(3)=4$, we can just show that $\displaystyle f'(\xi)=1$, but not $\displaystyle f(x)=x+1$
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    Re: differentiability and continuity problem

    Quote Originally Posted by maoro View Post
    #2 Assume $\displaystyle f(x)$ is continuous on $\displaystyle [a,b]$, and for any $\displaystyle x\in[a,b]$, there exists y\in[a,b] such that $\displaystyle |f(y)| \leq \frac{1}{2}|f(x)|$. Prove that there exists a point $\displaystyle c \in [a,b]$ such that $\displaystyle f(c)=0.$
    Hint: $\displaystyle g(x)=|h(x)|$ is continuous on $\displaystyle [a,b]$ .
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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 $\displaystyle f(3)=4$, we can just show that $\displaystyle f'(\xi)=1$, but not $\displaystyle f(x)=x+1$
    You can prove that $\displaystyle f'(x)=1$ for all $\displaystyle x\in (1,3)$ so $\displaystyle f(x)=x+C$ in $\displaystyle (1,3)$ 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 $\displaystyle f'(x)=1$ for all $\displaystyle x\in (1,3)$ so $\displaystyle f(x)=x+C$ in $\displaystyle (1,3)$ etc ...
    Is it because $\displaystyle f$ is differentiable, then if $\displaystyle f'(x)=1$, we get $\displaystyle f(x)=x+C$ ?
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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 $\displaystyle f$ is differentiable, then if $\displaystyle f'(x)=1$, we get $\displaystyle f(x)=x+C$ ?
    Right.
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    Re: differentiability and continuity problem

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

    $\displaystyle |f(c)|\leq \frac{1}{2}|f(x_1)|\leq \frac{1}{2^2}|f(x_2)|\leq...\leq\frac{1}{2^n}|f(x_ n)|$
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    Re: differentiability and continuity problem

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

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

    The question i want to ask is how can i prove $\displaystyle f(x)=x$ if i don't know $\displaystyle f$ 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 $\displaystyle f(x)=x$ if i don't know $\displaystyle f$ is differentiable ?
    You can't, there are infinite functions satisfying $\displaystyle f(0)=0$ and $\displaystyle f(4)=4$ .
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