# What is the concavity of a linear function?

Jun 10, 2018

Here's an approach...

#### Explanation:

Let's see...

A linear is in the form $f \left(x\right) = m x + b$ where $m$ is the slope, $x$ is the variable, and $b$ is the y-intercept. (You knew that!)

We can find the concavity of a function by finding its double derivative ($f ' ' \left(x\right)$) and where it is equal to zero.

Let's do it then!

$f \left(x\right) = m x + b$

$\implies f ' \left(x\right) = m \cdot 1 \cdot {x}^{1 - 1} + 0$

$\implies f ' \left(x\right) = m \cdot 1$

$\implies f ' \left(x\right) = m$

$\implies f ' ' \left(x\right) = 0$

So this tells us that linear functions have to curve at every given point.

Knowing that the graph of linear functions is a straight line, this does not make sense, does it?

Therefore, there is no point of concavity on the graphs of linear functions.