If I am finding the inflection points of a function using the first derivative graph, I recognize that it exists where the first derivative changes from increasing to decreasing or vice versa. In the example below, however, the graph is stationary from 3 to 4, so what is my inflection point (imagine that it is a first derivative graph, despite the label)? Is it 3, since that is where the graph of the derivative is no longer decreasing? Or is it 4, since that is where the graph of the derivative begins to increase?
Thanks for your help.
Best Answer
An inflection point is a point where the curve changes concavity, from up to down or from down to up. It is also a point where the tangent line crosses the curve. The tangent to a straight line doesn't cross the curve (it's concurrent with it.) So none of the values between $x=3$ to $x=4$ are inflection points because the curve is a straight line.
Is $x=3$ and inflection point? No. The concavity doesn't change from up to down or down to up. It changes from being there to not being there. The 2nd-derivative doesn't exist at that point, so it can't be an inflection point. It's a theorem that an inflection point is a point where the 2nd-derivative exists AND the curve changes concavity. Ditto for $x=4$.
The only inflection point for your function is at $x=6.$