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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?

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2 Answers 2

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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.

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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.

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