How to Calculate the Curvature of a Function

The curvature of a function shows how the function bends. Where the function has a maximum, it bends downwards, and we call it concave. Where the function has a minimum, the graph bends upwards, and we call it convex.

You find the curvature of a graph by looking at the sign chart of the second derivative $f^{″} (x)$ . It works like this:

Rule

Curvature of a Graph

The following relationship exists between the second derivative and the curvature of the graph.

$f^{″} (x) > 0 \Leftrightarrow convex, \cup$
$f^{″} (x) < 0 \Leftrightarrow concave, \cap$
$f^{″} (x) = 0 \Leftrightarrow$ inflection point, the graph is increasing or decreasing the fastest.

Note!

If the second derivative of the function is positive, the graph is positive (looks like it’s smiling). If the second derivative of the function is negative, the graph is negative (looks like it’s sad).

A function with its inflection point and curvature marked

Example 1

Describe the curvature of the graph given by

f (x) = x^{3} + x^{2} - 2 x

First, you differentiate the function twice like this:

\begin{array}{l} f^{'} (x) & = 3 x^{2} + 2 x - 2 \\ f^{″} (x) & = 6 x + 2 . \end{array}

By putting $f^{″} (x) = 0$ , you get $6 x + 2 = 0$ , which gives $x = - \frac{1}{3}$ . This is the $x$ -coordinate of the inflection point. As it is an inflection point, you know that the graph bends one way to the left of the inflection point and bends another way to the right of the inflection point. Thus, you only need to insert a value into $f^{″} (x)$ to check whether it is positive or negative. Just be sure to choose smart values, like $x = - 10$ and $x = 10$ . You then find that

\begin{array}{l} f^{″} (- 10) & = 6 (- 10) + 2 = - 58 \\ f^{″} (10) & = 6 (10) + 2 = 62 \end{array}

As $f^{″} (- 10) = - 58 < 0$ , the graph is concave in the interval $(- \infty, - \frac{1}{3})$ . As $f^{″} (10) = 62 > 0$ , the graph is convex in the interval $(- \frac{1}{3}, \infty)$ . It’s good to remember that $f^{″} (x)$ is the derivative of $f^{'} (x)$ , and that $f^{″} (x)$ is the second derivative of $f (x)$ . This means that $f^{″} (x)$ and $f^{'} (x)$ relate to each other in the same way that $f^{'} (x)$ and $f (x)$ relate to one another.

Rule

The Relationship Between $f (x)$ , $f^{'} (x)$ and $f^{″} (x)$

You have a solid line in the sign chart when:

1.: $f^{″} (x) > 0$ and when $f^{″} (x)$ is above the $x$ -axis $\Leftrightarrow f (x)$ is convex.
2.: $f^{'} (x) > 0$ and when $f^{'} (x)$ is above the $x$ -axis $\Leftrightarrow f (x)$ increases.
3.: $f (x) > 0$ and when $f (x)$ is above the $x$ -axis $\Leftrightarrow f (x)$ is positive.

You have a dotted line when:

1.: $f^{″} (x) < 0$ and when $f^{″} (x)$ is below the $x$ -axis $\Leftrightarrow f (x)$ is concave.
2.: $f^{'} (x) < 0$ and when $f^{'} (x)$ is below the $x$ -axis $\Leftrightarrow f (x)$ is decreasing.
3.: $f (x) < 0$ and when $f (x)$ is below the $x$ -axis $\Leftrightarrow f (x)$ is negative.

The relationship between f(x), f’(x) and f”(x)

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What Are Stationary Points of a Function?

What Are Inflection Points of a Function?

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Curvature of a Graph

Note!

The Relationship Between f(x), f′(x) and f″ ⁡(x)

The Relationship Between $f (x)$ , $f^{'} (x)$ and $f^{″} (x)$