1. ## Injective

Happy Thanksgiving to those of you in the States.

I have a problem:

Suppose that a function f [a,b] to R is continuous such that f attains its minimum or maximum on [a,b] at a point c such that
a<c<b.
Prove that f cannot be injective.

How would you work through this proof?

2. Can you tell us why you need three accounts?

RonL

3. Originally Posted by OntarioStud
Happy Thanksgiving to those of you in the States.

I have a problem:

Suppose that a function f [a,b] to R is continuous such that f attains its minimum or maximum on [a,b] at a point c such that
a<c<b.
Prove that f cannot be injective.

How would you work through this proof?
I am going to make this proof as if it was a maximum point the minimum point is anagolous.

Assume,
$\displaystyle f:[a,b]\to \mathbb{R}$ is an injective map.
Also that the function has a maximum $\displaystyle c\in (a,b)$.
$\displaystyle [a,c],[c,b]$
Without lose of generality assume $\displaystyle f(a)\geq f(b)$. Pick any number $\displaystyle d$ such that $\displaystyle f(b) \leq d< f(c)$
$\displaystyle f(a)\leq d< f(c)$
By the intermediate value theorem there is a point $\displaystyle m_{1,2}$ such that $\displaystyle f(m_1)=d$ on $\displaystyle [a,c]$ and $\displaystyle f(m_2)=d$ on $\displaystyle [c,b]$. Thus, $\displaystyle f(m_1)=f(m_2)$, and if $\displaystyle m_1\not = m_2$ the function fails to be injective. The only way it can is when $\displaystyle m_1=m_2=c$ but that is not possible.