Intervals of Concavity for Business Calculus

with sample problems

Intro

In business calculus, concavity is a word used to describe the shape of a curve. In general, a curve can be either concave up or concave down.


Concave Up

First, let's figure out how concave up graphs look.

concave up graph

Learn the concept

A concave up graph is a curve that "opens upward", meaning it resembles the shape $\cup$

You can think of the concave up graph as being able to "hold water", as it resembles the bottom of a cup.

For example, the graph of the function $y=x^2+2$ results in a concave up curve. To view the graph of this function, click here. Notice that the graph opens "up".


Concave Down

The opposite of concave up graphs, concave down graphs point in the opposite direction.

concave down graph

Learn the concept

On the other hand, a concave down curve is a curve that "opens downward", meaning it resembles the shape $\cap$

So, a concave down graph is the inverse of a concave up graph. Using the same analogy, unlike the concave up graph, the concave down graph does NOT "hold water", as the water within it would fall down, because it resembles the top part of a cap.

For example, the graph of the function $y=-3x^2+5$ results in a concave down curve. To view the graph, click here. Notice this graph opens "down".


Both Concave Up & Down

These two examples are always either concave up or concave down. However, a function can be concave up for certain intervals, and concave down for other intervals. This means that the graph can open up, then down, then up, then down, and so forth.

The perfect example of this is the graph of $y=sin(x)$. Click here to view the graph for this function. As you can see, the graph opens downward, then upward, then downward again, then upward, etc.

concavity equals zero

A Little Tip

Just as functions can be concave up for some intervals and concave down for others, a function can also not be concave at all. If the second derivative of the function equals $0$ for an interval, then the function does not have concavity in that interval.


Concavity Using Derivatives

You can easily find whether a function is concave up or down in an interval based on the sign of the second derivative of the function.

finding concavity with business calculus

Learn the concept

In words: If the second derivative of a function is positive for an interval, then the function is concave up on that interval. Otherwise, if the second derivative is negative for an interval, then the function is concave down at that point.

In math notation: If $f''(x) > 0$ for $[a,b]$, then $f(x)$ is concave up on $[a,b]$. Otherwise, if $f''(x) < 0$ for $[a,b$], then $f(x)$ is concave down on $[a,b]$

The concept is very similar to that of finding intervals of increase and decrease. We still set a derivative equal to $0$, and we still plug in values left and right of the zeroes to check the signs of the derivatives in those intervals.

The main difference is that instead of working with the first derivative to find intervals of increase and decrease, we work with the second derivative to find intervals of concavity.


How to Apply

In business calculus, you will be asked to find intervals of concavity for graphs. In other words, this means that you need to find for which intervals a graph is concave up and for which others a graph is concave down.

In general, concavity can only change where the second derivative has a zero, or where it is undefined. Here are the steps to determine concavity for $f(x)$:

  • Step 1: Find the second derivative of the original function
  • Step 2: If the second derivative only has a numerator (it is not a fraction), set the function equal to $0$
  • Step 3: Find the numbers that make the numerator equal to $0$ by solving for $x$ in Step 2
  • Step 4: Test one number to the left (less than) and one to the right (greater than) of each zero found in order to find the concavity on both sides
  • Step 5: If the second derivative has a denominator, repeat Steps 2, 3 & 4 for the denominator
  • Step 6: Based on all the zeros you found (both numerator and denominator) and the signs left and right, write out each interval and its concavity

While this might seem like too many steps, remember the big picture:

steps for concavity

The big picture

To find the intervals of concavity, you need to find the second derivative of the function, determine the $x$ values that make the function equal to $0$ (numerator) and undefined (denominator), and plug in values to the left and to the right of these $x$ values, and look at the sign of the results:

$+ \ \rightarrow$ interval is concave up

$- \ \rightarrow$ interval is concave down


Sample Problem

Given the function $f(x)=5x^3+30x^2$

Question 1Determine where this function is concave up and concave down

The first step in determining concavity is calculating the second derivative of $f(x)$. So, we differentiate it twice,

$$f'(x)=15x^2+60x$$

$$f''(x)=30x+60$$

Now that we have the second derivative, we want to find concavity at all points of this function. The function can either be always concave up, always concave down, or both concave up and down for different intervals.

We set the second derivative equal to $0$, and solve for $x$

$$30x+60=0$$

First, we subtract $60$ from both sides,

$$30x=-60$$

And then we divide by $30$ on both sides,

$$x=-2$$

This means that this function has a zero at $x=-2$. Therefore, we need to test for concavity to both the left and right of $-2$. Let's pick $-5$ and $1$ for left and right values, respectively. We build a table to help us calculate the second derivatives at these values:

$x$ $f''(x)$ $+$ or $-$
$-5$ $30(-5)+60=-90$ $-$
$1$ $30(1)+60=90$ $+$

As per our table, when $x=-5$ (left of the zero), the second derivative is negative. Also, when $x=1$ (right of the zero), the second derivative is positive. That gives us our final answer:

$in \ (-\infty,-2) \ \rightarrow \ f(x) \ is \ concave \ down$

$in \ (-2,+\infty) \ \rightarrow \ f(x) \ is \ concave \ up$

By the way, an inflection point is a graph where the graph changes concavity. Therefore, there is an inflection point at $x=-2$.

Lesson review

  • Graphs of curves can either be concave up or concave down
  • Concave up graphs open upward, and have the shape $\cup$
  • Concave down graphs open downward, with the shape $\cap$
  • To determine the concavity of a graph, find the second derivative of the given function and find the values that make it $0$ or undefined. Then check for the sign of the second derivative in all intervals
  • If $f''(x) > 0$, the graph is concave up on the interval. Else, if $f''(x)<0$, the graph is concave down on the interval

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