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You can use sign charts to analyze the behavior of a function. They help you find maxima, minima and saddle points. Here, it’s important to keep your head in the game. You’re looking to say something about the function f(x) based on its derivative f′(x). This is how you do it:

Rule

Drawing the Sign Chart of the Derivative

  • Find the derivative of f(x). Draw the sign chart of the differentiated function f′(x). You’re going to find out where this function is above and below the x-axis.

  • Mark which x-values give you positive y-values (solid line) and which ones give negative y-values (dashed line).

Rule

The Connection Between f(x) and f′(x)

It turns out that there is a clear connection between f(x) and f′(x):

  • When f′(x) is positive (above the x-axis), f(x) increases.

  • When f′(x) is negative (below the x-axis), f(x) decreases.

  • When f′(x) is zero (on the x-axis), f(x) has a maximum, minimum or saddle point.

Rule

Determine Stationary Points

1.
If the zero of f′(x) lies between two solid lines or between two dashed lines, then f(x) has a saddle point—see Figure (a).
2.
If the zero has a solid line on the left and a dashed line on the right, then f increases before the point and decreases after. That means the point is a maximum—see Figure (b).
3.
If the zero has a dashed line on the left and a solid line on the right, then f decreases before the point and increases after. That means the point is a minimum—see Figure (c).

Saddle point of an increasing function

(a)

Maximum of a function

(b)

Minimum of a function

(c)

Example 1

You have a cubic function f(x)=−23x3−x2+4x. Find the maxima and minima of f(x).

1.
First, you find the derivative of f(x):
f′(x)=−2x2−2x+4
2.
You can factorize this expression according to the formula
ax2+bx+c=a(x−x1)(x−x2)

where x1 and x2 are the solutions of ax2+bx+c=0. That means the factorized expression for the derivative is

f′(x)=−2(x2+x−2)=−2(x+2)(x−1)

f′(x)=−2(x2+x−2)=−2(x+2)(x−1)

3.
Then the sign chart looks like this:

Example of a sign chart of the derivative

Example of a sign chart of the derivative

4.
Now you have to find out which points are maxima and which are minima. The sign chart tells you that the function decreases until x=−2, increases for a bit until x=1, and then decreases again. That means x=−2 is a minimum and x=1 is a maximum.

Now you need to find the corresponding y-values. To do so, just insert the x-values you found into the function

f(x)=−23x3+x2+4x

That gives you:

Minimum point:

(xmin⁡,f(xmin⁡))=(−2,f(−2))=(2−,−203)

because

f(−2)=−23(−2)3−(−2)2+4(−2)=163+4−8=16−12−243=−203

f(−2)=−23(−2)3−(−2)2+4(−2)=163+4−8=16−12−243=−203

Maximum point:

(xmax⁡,f(xmax⁡))=(1,f(1))=(1,73)

because

f(1)=−23(1)3−(1)2+4(1)=−23−1+4=−23−33+123=73.

Example of graph illustrated using sign chart

Example of graph illustrated using sign chart

This picture fits with what we found through our calculations. The minimum is at (−2,−203) and the maximum is at (1,73). You can see that the graph decreases until it reaches the minimum, because the derivative of the function is negative and its sign chart has a dashed line.

Between the minimum and the maximum the graph increases, because the derivative of the function is positive and its sign chart has a solid line.

From the maximum and onward the graph decreases again, because the derivative of the function is negative and its sign chart has a dashed line.

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