# Convex and Concave Functions and Inflection Points

**Convex** and **concave** are words that we use to describe the shape or curvature of a curve. Recall from classifying stationary points (see Stationary Points page) that we can find the second derivative of a function by differentiating twice. We also require the second derivative to see if a function is concave or convex at a particular point. A **point of inflection** is where a curve changes from being concave to being convex or vice versa.

## Convex and Concave Functions

Recall firstly that solving $f'(x)=0$ for $x$ will identify the locations of the stationary points of the curve $y=f(x)$. Secondly, finding the value of $f”(x)$ at these stationary points allows us to classify the stationary points. That is if $f”(x)<0$, it is a maximum and if $f”(x)>0$, it is a minimum.

It is, however, possible to find the value of $f^{”}(x)$ at any point on a curve, not necessarily stationary points. If at any point on a curve, the second derivative is negative, we say that the curve is **concave** at that point. Conversely, if the second derivative is positive at any point, we say that the curve is **convex** at that point. It follows that there is an interval around a maximum that is concave and an interval around a minimum that is convex. See Example 1. The point where a curve changes from being concave to convex or vice versa is known as an inflection point.

## Points of Inflection

Consider the graph of $y=x^3$. The derivative is $\frac{dy}{dx}=3x^2$ and the second derivative is $\frac{d^2y}{dx^2}=6x$. The second deriative is 0 when $x=0$, it is positive when $x>0$ and negative when $x<0$. It follows that the point $(0,0)$ is an inflection point. Also, the curve is concave when $x<0$ and convex when $x>0$. A point of inflection is where a curve goes from being concave to convex or vice versa. This means that the second derivative changes sign. For the curve of $y=x^3$ the point $(0,0)$ is a stationary point as well as a point of inflection. However, an inflection point doesn’t have to be a stationary point as we can see in the section above. Also see Example 2. It is the fact that $f”(x)=0$ and $f”(x)$ changing sign that makes a given point one of inflection.

## Convex, Concave and Inflection Examples

Determine whether the curve of $y=x^2-e^x$ is convex or concave at the points with $x$-coordinates $x=0$ and $x=1$.

Differentiating we obtain $\frac{dy}{dx}=2x-e^x$ and $\frac{d^2y}{dx^2}=2-e^x$. At $x=0$ the second derivative is $\frac{d^2y}{dx^2}=2-1=1$. Since this is positive the graph of $y=x^2-e^x$ is convex at $x=0$. At $x=1$ the second derivative is $\frac{d^2y}{dx^2}=2-e^1\approx-0.718$. Since this is negative the curve graph of $y=x^2-e^x$ is concave at $x=1$.

Determine the the inflection point of the curve of $f(x)=x^2\ln(x)-\frac{5}{2}x^2$. Hence, determine the concave and convex regions. Is the point of inflection also a stationary point?

Differentiating using the product rule gives $f'(x)=2x\ln(x)+\frac{x^2}{x}-5x=2x\ln(x)-4x$. Differentiating again gives $f”(x)=2\ln(x)+2-4=2\ln(x)-2$. Solving $f”(x)=0$ gives $x=e$ (see more on solving log equations). It follows that $f”(x)<0$ for $x<e$ and $f”(x)>0$ for $x>0$. Hence, the curve is concave for $x<e$ and convex for $x>e$. It follows that there is a point of inflection at $x=e$. The corresponding $y$-coordinate is $f\left(e\right)=e^2\ln\left(e\right)-\frac{5}{2}e^2=-\frac{3}{2}e^2$. Hence, the point of inflection is at $\left(e,-\frac{3}{2}e^2\right)$. This point is not a stationary point since the derivative at this point is $f'(e)=2e\ln(e)-4e=-2e\ne 0$.