# The Second Derivative and the Shape of a Function f(x)

Site: | Saylor Academy |

Course: | MA005: Calculus I |

Book: | The Second Derivative and the Shape of a Function f(x) |

Printed by: | Guest user |

Date: | Tuesday, August 6, 2024, 12:33 PM |

## Description

Read this section to learn how the second derivative is used to determine the shape of functions. Work through practice problems 1-9.

## Introduction

The first derivative of a function gives information about the shape of the function, so the second derivative of a function gives information about the shape of the first derivative and about the shape of the function. In this section we investigate how to use the second derivative and the shape of the first derivative to reach conclusions about the shape of the function. The first derivative tells us whether the graph of is increasing or decreasing. The second derivative will tell us about the "concavity" of , whether is curving upward or downward.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.4-Second-Derivative.pdf

This work is licensed under a Creative Commons Attribution 3.0 License.

## Concavity

Graphically, a function is concave up if its graph is curved with the opening upward (Fig. 1a). Similarly, a function is concave down if its graph opens downward (Fig. 1b). The concavity of a function can be important in applied problems and can even affect billion-dollar decisions.

**Fig. 1**

An Epidemic: Suppose an epidemic has started, and you, as a member of congress, must decide whether the current methods are effectively fighting the spread of the disease or whether more drastic measures and more money are needed. In Fig. 2, is the number of people who have the disease at time , and two different situations are shown. In both and , the number of people with the disease, now), and the rate at which new people are getting sick, (now), are the same. The difference in the two situations is the concavity of , and that difference in concavity might have a big effect on your decision. In , is concave down at "now", and it appears that the current methods are starting to bring the epidemic under control. In , is concave up, and it appears that the epidemic is still out of control.

**Fig. 2**

Usually it is easy to determine the concavity of a function by examining its graph, but we also need a definition which does not require that we have a graph of the function, a definition we can apply to a function described by a formula without having
to graph the function.

**Definition:** Let be a differentiable function.

is **concave up** at a if the graph of is above the tangent line to for all close

is concave down at if the graph of is below the tangent line to for all close

Fig. 3 shows the concavity of a function at several points. The next theorem gives an easily applied test for the concavity of a function given by a formula.

**Fig. 3**

**The Second Derivative Condition for Concavity**

(a) If on an interval , then is increasing on and is concave up
on .

(b) If on an interval , then is decreasing on and is concave down on .

(c) If , then may be concave up or concave down or neither at .

Proof: (a) Assume that for all in , and let a be any point in . We want to show that is concave up at so we need to prove that the graph of
(Fig. 4) is above the tangent line to at : for close to .

**Fig. 4**

Assume that is in , and apply the Mean Value Theorem to on the interval from to . Then there is a number between and so that

Since between and , we know from the Second Shape Theorem that is increasing between and . We will consider two cases: and .

If , then and is in the interval so . Since is increasing, implies that . Multiplying each side
of the inequality by the positive amount , we get that .
Adding to each side of this last inequality, we have .

If , then and is in the interval so . Since is increasing, a implies that . Multiplying each side
of the inequality by the **negative** amount , we get that
and .

In each case we get that the function is above the tangent line . The proof of is similar.

(c) Let , and (Fig.5). The second derivative of each of these functions is zero at , and at they all have the same tangent line: , the -axis. However, at they all have
different concavity: is concave up, is concave down, and is neither concave up nor down.

**Fig. 5**

**Practice 1:** Use the graph of in Fig. 6 to finish filling in the table with "" for positive, "" for negative" or "".

**Fig. 6**

## Feeling the Second Derivative

Earlier we saw that if a function represents the position of a car at time , then is the velocity and is the acceleration of the car at the instant .

If we are driving along a straight, smooth road, then what we feel is the acceleration of the car:

a large positive acceleration feels like a "push" **toward the back** of the car,

a large negative acceleration (a deceleration) feels like a "push" **toward the front** of the car, and an acceleration of for a period
of time means the velocity is constant and we do not feel pushed in either direction.

On a moving vehicle it is possible to measure these "pushes", the acceleration, and from that information to Ä‘etermine the velocity of the vehicle, and from the velocity information to determine the position. Inertial guidance systems in airplanes use
this tactic: they measure front-back, left-right and up-down acceleration several times a second and then calculate the position of the plane. They also use computers to keep track of time and the rotation of the earth under the plane. After all,
in 6 hours the earth has made a quarter of a revolution, and Dallas has rotated more than 5000 miles!

**Example 1:** The upward acceleration of a rocket was for the first 6 seconds of flight, . The velocity of the rocket at was and the height of the rocket above the ground at was . Find a formula for the height of the rocket at time and determine the height at seconds.

Solution: so for some constant . We also know so and . Therefore,

Similarly, so for some constant . We know that so and . Then .

## f'' and Extreme Values

The concavity of a function can also help us determine whether a critical point is a maximum or minimum or neither. For example, if a point is at the bottom of a concave up function (Fig. 7), then the point is a minimum.

**Fig. 7**

**The Second Derivative Test for Extremes:**

(a) If and then is concave down and has a local maximum at .

(b)
If and then is concave up and has a local minimum at .

(c) If and
then may have a local maximum, a minimum or neither at .

Proof: (a) Assume that . If then is concave down at so the graph of will be below the tangent line
for values of near . The tangent line, however, is given by , so if is close to then and has a local maximum at . The proof
of (b) for a local minimum of is similar.

(c) If and , then we cannot immediately conclude anything about local maximums or minimums of at . The functions , and
all have their first and second derivatives equal to zero at , but has a local minimum at has a local maximum at , and has neither a local maximum nor a local minimum at .

The Second Derivative Test for Extremes is very useful when is easy to calculate and evaluate. Sometimes, however, the First Derivative Test or simply a graph of the function is an easier way to determine if we have a local
maximum or a local minimum â€“ it depends on the function and on which tools you have available to help you.

**Practice 2:** has critical numbers and . Use the Second Derivative Test for Extremes to determine whether and
are maximums or minimums or neither.

## Inflection Points

**Definition:** An **inflection point** is a point on the graph of a function where the concavity of the function changes, from concave up to down or from concave down to up.

**Practice 3:** Which of the labelled points in Fig. 8 are inflection points?

**Fig. 8**

To find the inflection points of a function we can use the second derivative of the function. If , then the graph of is concave up at the point so is not an inflection point. Similarly,
if , then the graph of is concave down at the point and the point is not an inflection point. The only points left which can possibly be inflection points are the
places where is or undefined ( is not differentiable). To find the inflection points of a function we only need to check the points where is or
undefined. If or is undefined, then the point **may** or **may not** be an inflection point â€“ we would need to check the concavity
of on each side of . The functions in the next example illustrate what can happen.

**Example 2:** Let and (Fig. 9). For which of these functions is the point an inflection point?

**Fig. 9**

Solution: Graphically, it is clear that the concavity of and changes at , so is an inflection point for and . The function
is concave up everywhere so is not an inflection point of .

If , then and . The only point at which or is undefined ( is not differentiable) is at . If , then
so is concave down. If , then so is concave up. At the concavity changes so the point is an inflection point of .

If , then and . The only point at which or is undefined
is at . If , then so is concave up. If , then so is also concave up. At the concavity **does not change** so the point
is **not an inflection point** of .

If , then and is not defined
if , but (negative number) and (positive number) so changes concavity at and is an inflection point of .

**Practice 4:** Find the inflection points of .

**Example 3:** Sketch graph of a function with , and an inflection point at . Solution: Two solutions are given in Fig. 10.

**Fig. 10**

## Practice Answers

**Practice 1:** See Fig. 6.

**Fig. 6**

so is concave down at the critical value so is a rel. max.

so is concave up at the critical value so is a rel. min.

**Fig. 18**

**Practice 3:** The points labeled and in Fig. 8 are inflection points.

**Practice 4: **

The only candidates to be Inflection Points are and .

If , then (neg)(neg) is positive.

If , then (pos )(neg) is negative.

If , then
(pos)(pos) is positive.

changes concavity at and so **and** **are Inflection Points**.

**Fig. 19**