Question

How can I tell on a graph if a point is an inflection point?

Answer 1

Do you mean visually?

Answer 2

I'm solving a problem and I found that f'(x) = (-1-x^2) / (1 + x^2)^2. So
f''(x) = 2x / (x^2 +1)^2.

Any roots for f''(x) would indicate inflection points right? How do I test if the slope rate increases or decreases to the left or right of the point?

Answer 3

That is correct. Inflection points are critical points of the first derivative, or where the second derivative is zero. I don't really understand your last question about the slope rates...

Answer 4

Ok so concave up means that the slope is gradually increasing, and concave down means the slope is gradually decreasing. So for instance if I take the inflection point and input a number to the left of it into f''(x), will it give me the concavity to the left of the inflection point?

Answer 5

That's right! Notice that at zero, the second derivative is negative to the left and positive to the right. That means that right at x = 0, there is an inflection point.

Answer 6

Ok thanks! So basically finding a root of f''(x) alone will not tell you if it is an inflection point or not, right? It needs to have a change of value from the left to the right of the root too?

Answer 7

Correct. It is possible to have a situation where the second derivative has a root, but the second derivative doesn't change sign. That is called an undulation point (according to wikipedia). An example would be the graph of x^4.

Answer 8

Thanks a lot!