Thread: Continuous function, maximum and minimum

1. Continuous function, maximum and minimum

I'm exercising for a test I have in two days, and I came across this question from previous exams:

Let f : (0,1) ---> R be a continuous function. It is given that f does not have maxima nor minima, and that f(0.5) = f(3/4).
Show that f has a local minimum point in (0,1).

Any ideas?

2. Re: Continuous function, maximum and minimum

Originally Posted by loui1410
Let f : (0,1) ---> R be a continuous function. It is given that f does not have maxima nor minima, and that f(0.5) = f(3/4). Show that f has a local minimum point in (0,1).
Here are some hints. But it still takes some work.
The high-point/low-point theorem says that any continuous function on a closed interval has a maximum and a minimum.
$\displaystyle [0.5,0,75]$ is a closed interval. So?
But you know that f does not have maxima nor minima on $\displaystyle (0,1)$.
Thus what can you conclude?

3. Re: Continuous function, maximum and minimum

That f is a constant function? But anyhow how am I supposed to prove that it has a local minimum point when it says that it has no maxima nor minima? Do they actually mean that it has no global maxima or minima?

4. Re: Continuous function, maximum and minimum

Originally Posted by loui1410
That f is a constant function?
If the function is constant then it has both a global maxima and minima.

Originally Posted by loui1410
Do they actually mean that it has no global maxima or minima?
Of course that is exactly what it means. There is a $\displaystyle p\in [0.5,0.75]$ such $\displaystyle f(p)$ is a minimum on that interval.

If $\displaystyle f(p)\ne f(0.5)$ you are done.

BUT if $\displaystyle f(p)=f(0,5)$ there is more work to be done.