Maximum, minimum, and continuity

[Karol]

Homework Statement

Theorem 3 that i will give in the attempt at a solution talks about a closed interval, here it's open

Homework Equations

Continuity:
$$\vert x-c \vert < \delta~\Rightarrow~\vert f(x)-f(c) \vert < \epsilon$$
$$\delta=\delta(c,\epsilon)$$

The Attempt at a Solution

 

[Mark44]

Homework Statement

View attachment 204727
Theorem 3 that i will give in the attempt at a solution talks about a closed interval, here it's open

Homework Equations

Continuity:
$$\vert x-c \vert < \delta~\Rightarrow~\vert f(x)-f(c) \vert < \epsilon$$
$$\delta=\delta(c,\epsilon)$$

The Attempt at a Solution

View attachment 204728

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.

 

[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

 

[dgambh]

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

 

[LCKurtz]

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

 

[Ray Vickson]

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

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

 

[pasmith]

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

 

[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

 

[Mark44]

Colored text added.

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

 

[Karol]

So is it the answer?

 

[Ray Vickson]

So is it the answer?

Already answered!

 

[Karol]

Thank you very much Ray, Mark, pasmith, LCKurtz and dgambh