# How do you determine the intervals where f(x)=3x-4 is concave up or down?

Jun 30, 2018

$f \left(x\right) = 3 x - 4$ is never concave up or concave down.

#### Explanation:

By definition, a function $f \left(x\right)$ is concave up when $f ' ' \left(x\right) > 0$, and it is concave down when $f ' ' \left(x\right) < 0$.

Let $f \left(x\right) = 3 x - 4$.

$f ' \left(x\right) = 3$

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

Here, we notice that the second derivative is never greater than or less than 0, which means $f \left(x\right) = 3 x - 4$ is never concave up or concave down.

Jun 30, 2018

Neither- point of inflection

#### Explanation:

When we want to determine if a function is concave up or concave down, we want to analyze the function's second derivatives

$f ' \left(x\right) = 3$

$f ' ' \left(x\right) = 0$ (Derivative of a constant is zero)

We have three possible scenarios:

1. $f ' ' \left(x\right) > 0 \implies$Function is concave up

2. $f ' ' \left(x\right) < 0 \implies$Function is concave down

3. $f ' ' \left(x\right) = 0 \implies$Point of inflection (neither concave up or down)

We see that our second derivative of $f \left(x\right)$ is zero, which means we are in scenario three:

$f \left(x\right)$ is neither concave up nor down...we have a point of inflection.

Hope this helps!