Question

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

Answer 1

`f(x) = 3x-4` is never concave up or concave down.

Explanation:

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

Let `f(x) = 3x - 4`.

`f'(x) = 3`

`f''(x) = 0`

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

Answer 2

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'(x)=3`

`f''(x)=0` (Derivative of a constant is zero)

We have three possible scenarios:

1. `f''(x)>0=>`Function is concave up

2. `f''(x)<0=>`Function is concave down

3. `f''(x)=0=>`Point of inflection (neither concave up or down)

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

`f(x)` is neither concave up nor down...we have a point of inflection.

Hope this helps!