# Maximum, minimum, and continuity

• Karol
In summary, the problem is asking for the maximum or minimum of a function that takes in a closed interval, and the answer is that it is the supremum.
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:
$$\vert x-c \vert < \delta~\Rightarrow~\vert f(x)-f(c) \vert < \epsilon$$
$$\delta=\delta(c,\epsilon)$$

Karol said:

## 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.

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

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

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

Karol said:
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!

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

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

Karol said:
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 said:

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

## 1. What is the definition of a maximum and minimum value?

A maximum value is the largest value that a function or data set can take on within a given range. A minimum value is the smallest value that a function or data set can take on within a given range.

## 2. How do you find the maximum and minimum values of a function?

To find the maximum and minimum values of a function, you can use calculus techniques such as taking the derivative and setting it equal to 0 to find critical points. Then, you can evaluate the function at these points to determine the maximum and minimum values.

## 3. What is the difference between absolute and local maximum and minimum values?

An absolute maximum or minimum value is the largest or smallest value of a function over its entire domain. A local maximum or minimum value is the largest or smallest value of a function within a specific interval or region.

## 4. What does it mean for a function to be continuous?

A function is continuous if it has no breaks or jumps in its graph. This means that the function can be drawn without lifting the pen from the paper, and there are no sudden changes in the output values as the input values change.

## 5. How do you determine if a function is continuous at a specific point?

A function is continuous at a specific point if the limit of the function as the input approaches that point is equal to the output value at that point. In other words, the left and right limits of the function at that point must exist and be equal to the function value at that point.

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