Extrema and concavity

A look at first and second derivative tests.

Contents

• Higher derivatives
• Purpose
• Extrema
• Concavity
• First derivative test
• Second derivative test
• Global extrema
• Points of inflexion
• Tips

Higher derivatives

Derivative functions can also be differentiated. The second-derivative of $f$ is

At HL, there is also the notation

for the $n$th derivative.

Purpose

In this course, the purposes of identifying extrema are primarily

1. Curve sketching
2. Identifying intervals of increasing and decreasing
3. Optimization problems
4. Kinetics problems

The purpose of identifying concavity is mainly curve sketching-related.

So extrema and concavity are not only goals, but also tools.

Extrema

If the function is not defined over all reals, then these endpoints are automatically local extrema, provided that the function is defined on these endpoints.

Or, if the function is piece-wise, then where pieces are not continuous, such endpoints are also local extrema.

Otherwise, $x = x_0$ is a critical point, or candidate for a local maximum or local minimum if

1. $f^\prime(x_0) = 0$, OR
2. $f$ is continuous at $x = x_0$ but $f^\prime(x_0)$ is undefined (not differentiable).

Both tests are applicable when $f^\prime(x_0) = 0$; only the first derivative test is applicable when $f^\prime(x_0)$ is undefined.

Passing either the first or second derivative test means an extremum on $x = x_0$. Failing either test (as long as it is applicable) means there is no extremum on $x = x_0$.

The global maximum (minimum) is the greatest (least) of the local maxima (minima).

Concavity

Informally, a portion of the graph looking like $\cup$ is concave up, while ones looking like $\cap$ is concave down. The point where a graph changes concavity is known as the point of inflexion.

First derivative test

With critical point $x = x_0$, pick $a$ and $b$ such that $a < x_0 < b$. There cannot be other critical points in $[a, b]$. Try to make the algebra easy, if applying first derivative test by hand.

Second derivative test

The second derivative test requires the first derivative to be defined (and continuous).

Find $f^{\prime\prime}(x)$, and evaluate $f^{\prime\prime}(x_0)$.

Global extrema

Global max and min can be determined by comparing all the endpoint values and critical values, using $f(x)$.

Points of inflexion

A point of inflexion at $x = x_1$ requires

• $f^\prime(x_1)$ is defined
• $f^{\prime\prime}(x_1) = 0$
• no sign change around $f^\prime(x_1)$ or sign change around $f^{\prime\prime}(x_1)$.

Note that a POI is only a horizontal POI if the first derivative is also $0$. A POI need not satisfy either the first or second derivative tests.

Tips

1. Does the question care about all extrema or just ones on an open or closed interval?
2. Does question state that a max or min exist, or do you have to prove it’s a max or min?
3. Did you check the endpoints? Do you need to check them?
4. Does question want $x$ or $y$ or both?
5. Is it easier to find the second derivative or evaluate at values near the critical point?