# Maximum, minimum, and continuity

1. Jun 2, 2017

### Karol

1. The problem statement, all variables and given/known data

Theorem 3 that i will give in the attempt at a solution talks about a closed interval, here it's open
2. Relevant equations
Continuity:
$$\vert x-c \vert < \delta~\Rightarrow~\vert f(x)-f(c) \vert < \epsilon$$
$$\delta=\delta(c,\epsilon)$$

3. The attempt at a solution

2. Jun 2, 2017

### Staff: Mentor

Theorem 3 is no help here, since the interval for this problem is 0 < x < 1. The problem is fairly simple -- you shouldn't need to invoke a theorem to answer it.

3. Jun 2, 2017

### Karol

The derivative 2x is horizontal at 0, so there is a minimum or maximum. the fact that there isn't any other zero derivative proves there isn't another bending and it's rising, so the maximum is at 1

4. Jun 2, 2017

### dgambh

Remember $0 < x < 1$. Also you don't need to think this in terms of derivatives.

5. Jun 2, 2017

### LCKurtz

$x=0$ and $x=1$ are not in the domain of your function.

6. Jun 2, 2017

### Ray Vickson

The points x=0 and x=1 are not in the domain of the given function!

7. Jun 2, 2017

### pasmith

Hint: What is the distinction between a maximum and a supremum?

8. Jun 2, 2017

### Karol

1 is the supremum, x2 has no maximum.
0 is the infinum, x2 has no minimum
Is it the answer? it isn't part of the chapter, i learned about supremum here

9. Jun 2, 2017

### Staff: Mentor

10. Jun 2, 2017

### Karol

11. Jun 2, 2017