# Determining concavity and inflection points from a graph

I'm self-studying calculus by using Khan Academy, and I find myself consistently getting problems wrong where I'm asked to determine concavity from a graph or inflection points from a graph. Algebraically, I have no problem solving these problems: I can compute the second derivative, set that equal to $$0$$, and use text points within the resulting regions to check the sign of the second derivative. On a graph, it boils down to determining whether the slope of the second derivative is increasing from the function graph. It may be increasing, but at a "decreasing rate," or decreasing, but at a "decreasing rate," and so forth. I'm struggling to understand this graphically.

Does anyone have any tips on how I can get better at this?

Assuming your function $$f$$ is at least twice-differentiable, determining concavity from a graph boils down to the position of the graph relative to a tangent line. Given a point $$x_0$$, if the graph of $$f$$ lies above its tangent line near $$x_0$$ then it's concave up at $$x_0$$ (often called convex), and if it lies below then it's concave down at $$x_0$$ (sometimes just called concave). Having the slope of the tangent line increasing essentially forces the graph of $$f$$ to lie above the tangent near $$x_0$$, and similarly in the other case.
Intuitively, if the graph lies both above and below a tangent line near $$x_0$$, then $$x_0$$ will be an inflection point.
Given the function $$f$$, $$f$$ is concave down, so bent downwards on the interval $$I$$ if $$f'$$ is strictly monotonic decreasing on the interval $$I$$. $$f$$ is concave up on the interval $$I$$, so bent upwards, if $$f'$$ is strictly monotonic increasing on the interval $$I$$.
An inflection point occurs at a point where the function changes its concavity from concave up to concave down or concave down to concave up. At inflection points, $$f'$$ has extrema.
Thus, when given a graph of a function $$f$$, if on the interval $$I$$ the graph is bent upward, so the slope of $$f$$ is increasing, it is concave up, if the graph is bent downward, so the slope of $$f$$ is decreasing, it is concave down. You can find inflection points on a graph by finding the points where the curvature changes and $$f'$$ has an extremum.