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 $\displaystyle f(c)=0.$

Re: differentiability and continuity problem

Quote:

Originally Posted by

**maoro** #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)

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 $\displaystyle f(x)=x+1$

Re: differentiability and continuity problem

Quote:

Originally Posted by

**maoro** #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 $\displaystyle [a,b]$ .

Re: differentiability and continuity problem

Quote:

Originally Posted by

**maoro** I have just learned about Mean Value Theorem, is it similar to Lagrange Theorem?

Yes, it is another name.

Quote:

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** ...

Re: differentiability and continuity problem

Quote:

Originally Posted by

**FernandoRevilla** 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 $\displaystyle f(x)=x+C$ ?

Re: differentiability and continuity problem

Quote:

Originally Posted by

**maoro** Is it because $\displaystyle f$ is differentiable, then if $\displaystyle f'(x)=1$, we get $\displaystyle f(x)=x+C$ ?

Right.

Re: differentiability and continuity problem

Quote:

Originally Posted by

**FernandoRevilla** __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

$\displaystyle |f(c)|\leq \frac{1}{2}|f(x_1)|\leq \frac{1}{2^2}|f(x_2)|\leq...\leq\frac{1}{2^n}|f(x_ n)|$

Re: differentiability and continuity problem

Quote:

Originally Posted by

**maoro** $\displaystyle |f(c)|\leq \frac{1}{2}|f(x_1)|\leq \frac{1}{2^2}|f(x_2)|\leq...\leq\frac{1}{2^n}|f(x_ n)|$

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.

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 ?

Re: differentiability and continuity problem

Quote:

Originally Posted by

**maoro** 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 $\displaystyle f(4)=4$ .