# Show that affine functions are both concave and convex

• retspool
In summary, an affine function is a type of linear function that includes a constant term and can be written in the form f(x) = mx + b. To show that it is concave or convex, one can use the definition of concavity or convexity. The significance of affine functions being both concave and convex is that they have a unique global minimum and maximum value, making them useful in optimization problems. An example of an affine function is f(x) = 2x + 5. Affine functions and linear functions are related, with the main difference being the inclusion of a constant term in affine functions.
retspool

## Homework Statement

In my textbook, the author briefly makes a statement that affine functions are both concave and convex, how is that true? and how can it be proven?

## The Attempt at a Solution

Isn't an affine function a line?
Lines don't have any concavity.

This is exactly how the problem goes
let f(x) be a function in Rn.
prove that f(x) is both concave and convex if f(x) = cTx for some vector c

I thought that the function was a affine function, but i can't prove it

## 1. What is an affine function?

An affine function is a mathematical function that can be written in the form f(x) = mx + b, where m and b are constants and x is the independent variable. It is a type of linear function that includes a constant term.

## 2. How do you show that affine functions are concave and convex?

To show that an affine function is concave, you can use the definition of concavity which states that a function is concave if the line segment connecting any two points on the graph of the function lies entirely above the function. To show that an affine function is convex, you can use the definition of convexity which states that a function is convex if the line segment connecting any two points on the graph of the function lies entirely below the function.

## 3. What is the significance of affine functions being both concave and convex?

The fact that affine functions are both concave and convex means that they have a unique global minimum and maximum value. This makes them useful in optimization problems, as the optimal solution can be found by simply finding the point where the function reaches its minimum or maximum value.

## 4. Can you provide an example of an affine function?

One example of an affine function is f(x) = 2x + 5. This function has a slope of 2 and a y-intercept of 5, and can be graphed as a straight line.

## 5. How do affine functions relate to linear functions?

Affine functions and linear functions are closely related, as all affine functions are also linear functions. The main difference between the two is that affine functions include a constant term, while linear functions do not. This means that affine functions can have a y-intercept, while linear functions always pass through the origin.

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