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Concavity, Points of Inflection, and the Second Derivative Test

Concavity and Points of Inflection

Definitions:

If the graph of $f$ lies above all of its tangent lines on an open interval, the we say it is concave up on that interval.

If the graph of $f$ lies below all of its tangent lines on an open interval, then we say it is concave down on that interval.

A point, $P$, on a continuous curve $f(x)$ is an inflection point if $f$ changes concavity there.

When a curve is concave up, it is sort of bowl-shaped, and you can think it might hold water. When it is concave down, it is sort of upside-down-bowl-like, and water would run off of it.

How-to

The intervals of concavity can be found in the same way used to determine the intervals of increase/decrease, except that we use the second derivative instead of the first. In particular, since $(f')'=f''$, the intervals of increase/decrease for the first derivative will determine the concavity of $f$:

If possible, factor $f''$. If $f''$ is a quotient, factor the numerator and denominator (separately).

Find all critical numbers $x=s$ of $f'$. These are the points where $(f')'=0$ or $(f')'$ doesn't exist (i.e., the points where $f''=0$ or where $f''$ doesn't exist).

Draw a number line with tick marks at each critical number $s$.

For each interval in which the function $f$ is defined, find the sign of the second derivative $f''$.

If $f''(b) \gt 0$, then $f'$ is increasing on the interval containing $b$. This means that the slopes are increasing, so $f$ is concave up. Draw a right-side-up bowl over that interval on your number line. Similarly, if $f''(b) \lt 0$, draw an upside-down bowl.

That's it! You can now see the intervals where $f$ is concave up or down.

The Second Derivative Test

Theorem:

If $f''(x)>0$ for all $x$ in an open interval, then it is concave up on that interval.

If $f ''(x) < 0$ for all $x$ in an open interval, then it is concave down on that interval.

Second Derivative Test:

If $f'(c)=0$ and $f''(c) \gt0$, then there is a local minimum at $x=c$.

If $f'(c)=0$ and $f''(c) \lt 0$, then there is a local maximum at $x=c$.

If $f'(c)=0$ and $f''(c)=0$, or if $f''(c)$ doesn't exist, then the test is inconclusive. There might be a local maximum or minimum, or there might be a point of inflection.

The reasoning behind the test is simple: if $f''(c) \gt 0$, then $f'(x)$ is increasing near $x=c$. Since $f'(c)=0$, this means that $f'(x)$ used to be negative and is about to be positive. So the curve bottoms out at $x=c$ and then heads back up. The critical number $x=c$ is the bottom of the concave-up bowl. Likewise, if $f''(c) \lt 0$ and $f'(c)=0$, then $f'(x)$ is decreasing; it used to be positive and is about to be negative. The point $x=c$ is at the top of an upside-down bowl.

Inflection Points

An inflection point is a point where concavity changes sign from plus to minus or from minus to plus.

Example: Find the concavity of $f(x) = x^3 - 3x^2$.

Solution: Since $f'(x)=3x^2-6x=3x(x-2)$,our two critical points for $f$ are at $x=0$ and $x=2$. Meanwhile, $f''(x)=6x-6$, so the only critical point for $f'$ is at $x=1$. It's easy to see that $f''$ is negative for $x \lt 1$ and positive for $x \gt 1$, so our curve is concave down for $x \lt 1$ and concave up for $x \gt 1$, and there is a point of inflection at $x=1$.

As for the critical points, $f''(0)=-6 \lt 0$, so we have a local maximum at $x=0$. $f''(2)=6 > 0$, so we have a local minimum at $x=2$. These results agree with what we got from the first derivative test.

Visual Wrap-up

	$f'< 0$	$f'=0$	$f'> 0 $
$f''<0$
$f''=0$
$f''>0$

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits

Limit Laws and Computations

Continuity and the Intermediate Value Theorem

Limits at Infinity

Rates of Change

The Derivative Function

Basic Differentiation Rules

Product and Quotient Rules

Derivatives of Trig Functions

The Chain Rule

Implicit Differentiation

Derivatives of Logs

Derivatives in Science

Related Rates

Linear Approximation and Differentials

Differentiation Review

Absolute and Local Extrema

The Mean Value and other Theorems

$f$ vs. $f'$

Indeterminate Forms and L'Hospital's Rule

Optimization

Newton's Method

Anti-derivatives

The Area under a curve