Question

How do you find all points of inflection given `y=2x^2+4x+4`?

Answer 1

There are no points of inflection.

Explanation:

Points of inflection are places where the concavity of a function changes. Analytically, this happens when the sign of the second derivative `f''(x)` of a function changes.

We first find `f''(x)`:

`f(x) = 2x^2 + 4x + 4`

`f'(x) = 4x + 4`

`f''(x) = 4`

The second derivative is a constant value (4), meaning that the second derivative is defined always (it's 4), and it will never change sign. Thus, there are no points of inflection. This is demonstrated also by the graph, which is a "bowl up" parabola, and thus never changes from bring "bowl up" to "bowl down".