Question

Does the second derivative need to exist for there to be an inflection point?

Answer 1

what kind of equation has only a first derivative?

If you have a linear equation, such that f(x)=3x-4, the derivative is 3. Think about the graph of the derivative; inflection points are a switch in concavity. Can you have that without a second derivative?

Answer 2

y=3 Is an undefined line, so it wouldnt have an inflection points,

Answer 3

To find inflection points, you have to take the second derivative, so it DOES have to exist.