Introduction to Concavity

#Understanding Concavity

Concavity describes the direction in which a curve bends and is related to the second derivative of a function.

Key Concept

Concavity is determined by the sign of the second derivative, ${f}^{\text{'}\text{'}}(x)$ .

#Concave Up and Concave Down

A curve is:

Concave up if ${f}^{\text{'}\text{'}}(x) \ge 0$ for all values of $x$ in an interval.
- ${f}^{\text{'}}(x)$ is increasing in this interval.
- For example, the function $f(x) = x^2$ is concave up because ${f}^{\text{'}\text{'}}(x) = 2 \ge 0$ .
Concave down if ${f}^{\text{'}\text{'}}(x) \le 0$ for all values of $x$ in an interval.
- ${f}^{\text{'}}(x)$ is decreasing in this interval.
- For example, the function $f(x) = -x^2$ is concave down because ${f}^{\text{'}\text{'}}(x) = -2 \le 0$ .

Exam Tip

Remember: Concave up looks like a happy smiley face 🙂, and concave down looks like a sad smiley face ☹️.

#Points of Inflection

A point of inflection is where a graph changes concavity, from concave up to concave down or vice versa.

At a point of inflection, the second derivative is zero: ${f}^{\text{'}\text{'}}(x) = 0$ .
However, not every point where ${f}^{\text{'}\text{'}}(x) = 0$ is a point of inflection. The concavity must change.

For example, consider $f(x) = x^4$ :

${f}^{\text{'}\text{'}}(x) = 12x^2$
${f}^{\text{'}\text{'}}(0) = 0$ , but $x = 0$ is not a point of inflection because ${f}^{\text{'}\text{'}}(x) = 12x^2 > 0$ for all $x \ne 0$ .

#Exam Tips

Exam Tip

In an exam, always check the sign of the second derivative to determine concavity. Concave down implies ${f}^{\text{'}\text{'}}(x)$ is negative, and concave up implies ${f}^{\text{'}\text{'}}(x)$ is positive.

#Worked Example

The function $f$ is defined by:

$f(x) = \sin x, \quad 0 \le x \le 2\pi$

State the interval for which $f$ is concave down.

Solution:

A function is concave down when ${f}^{\text{'}\text{'}}(x)$ is negative.

$\begin{align*} {f}^{\text{'}}(x) &= \cos x \\ {f}^{\text{'}\text{'}}(x) &= -\sin x \end{align*}$

We need to solve: $-\sin x \le 0, \quad 0 \le x \le 2\pi$

The function is concave down where $-\sin x \le 0$ .

By inspecting the graph of $y = -\sin x$ for $0 \le x \le 2\pi$ , we see that the graph is less than zero on the interval $[0, \pi]$ .

Thus, the function $f(x)$ is concave down on the interval $[0, \pi]$ .

#Practice Questions

Practice Question

Determine the concavity of the function $f(x) = x^3 - 3x^2 + 2$ .

Practice Question

Find the points of inflection for the function $f(x) = \frac{1}{3}x^3 - x^2 + 2x - 1$ .

Practice Question

For the function $f(x) = e^x$ , identify the intervals where the function is concave up and concave down.

#Glossary

Concavity: The direction in which a curve bends.
Second Derivative: The derivative of the first derivative of a function, denoted ${f}^{\text{'}\text{'}}(x)$ .
Concave Up: When ${f}^{\text{'}\text{'}}(x) \ge 0$ in an interval.
Concave Down: When ${f}^{\text{'}\text{'}}(x) \le 0$ in an interval.
Point of Inflection: A point where the graph changes concavity.

#Summary and Key Takeaways

Concavity is determined by the second derivative of a function.
A function is concave up if ${f}^{\text{'}\text{'}}(x) \ge 0$ and concave down if ${f}^{\text{'}\text{'}}(x) \le 0$ .
Points of inflection occur where the second derivative is zero, and the concavity changes.
Use visual aids like graphs to better understand concavity and points of inflection.

#Real-World Applications

Concavity is crucial in various fields such as economics, physics, and engineering. For instance:

In economics, concavity can help determine the increasing or decreasing returns of a production function.
In physics, it helps in understanding the motion of objects under various forces.
In engineering, it aids in the design of curves and structures to ensure stability and efficiency.

Remember to practice these concepts regularly and apply them to different problems to master the topic of concavity!