What is concavity of a function?

Answer 1

Just as the slope of the tangent line to the graph of f at the point (x, f(x)) describes the behavior of the function, concavity describes the behavior of the slope. As x increases (graph goes from left to right), one of the following is true:

• Concavity is positive, so the slope slowly increases.
• Concavity is negative, so the slope slowly decreases.
• Concavity is equal to zero, so the slope is constant.

Again, remember that concavity directly affects the slope, NOT the function itself. I mean this in the sense that concavity affects slope affects function.

Mathematically speaking, you can find the concavity at a certain point by taking the derivative of the derivative of the function (accurately called the second derivative, f''). So, when you take the derivative of a function, you get the first derivative, f' (describing slope), and the derivative of that is the second derivative (describing the concavity).

Last but not least, here is a handy way to find the concavity of a function by looking at its graph:

• Concavity is positive when the graph turns up, like a smiling emoticon (look at a graph of f(x) = x2 for an example).

First observe that f'(x) = 2x.

We see that f' < 0 when x < 0 and f' > 0 when x > 0. So that the graph is decreasing on the negative real axis and the graph is increasing on the positive real axis.

Next observe that f''(x) = 2.

Thus, f'' > 0 at all points. Thus the graph is concave up everywhere.

Finally observe that the graph passes through the origin.

• Concavity is negative when the graph turns down, like a frowning emoticon (look at a graph of f(x) = -x2 for an example).

First observe that f'(x) = -2x.

We see that f' > 0 when x < 0 and f' < 0 when x > 0. So that the graph is increasing on the negative real axis and the graph is decreasing on the positive real axis.

Next observe that f''(x) = -2.

Thus, f'' < 0 at all points. Thus the graph is concave down everywhere.

Finally observe that the graph passes through the origin.

Look at the graph of f(x) = x3

First observe that f'(x) = 3x2.

Thus, f' ≥ 0 everywhere. The function is always increasing.

Next observe that f''(x) = 6x.

Thus, f'' < 0 when x < 0 and f'' > 0 when x > 0. So the graph is concave down on the negative real axis and concave up on the positive real axis.

Finally observe that the graph passes through the origin.

• Concavity is zero when the graph is linear OR at a point where it stops turning up and starts turning down, and vice versa.