Why Joining Points Need Special Attention
A piecewise-defined function may follow one rule to the left of a point and a different rule to the right. At the joining point, the graph can change direction sharply, forming a corner. Because the ordinary derivative may not exist there, classifying a possible local extremum requires looking at behavior on both sides.
The key question is not whether a derivative exists at the join. It is whether nearby function values are above or below the value at the join. One-sided derivatives and derivative signs can provide that information, but they must be interpreted carefully.
One-Sided Derivatives at a Join
Suppose \(f\) is defined at \(x=c\). The left-hand derivative, when it exists, is the limit of the difference quotient as \(x\) approaches \(c\) from below. The right-hand derivative is the corresponding limit as \(x\) approaches from above.
If these two finite values are different, then \(f'(c)\) does not exist. In a continuous piecewise-defined function, that unequal-slope join is a typical corner. The one-sided slopes describe how the graph approaches and leaves the point, but the sign of each slope must be read in relation to the side it describes.
When both one-sided derivatives are nonzero, their signs directly compare nearby function values with \(f(c)\). A positive left-hand derivative means nearby values to the left are less than \(f(c)\); a negative left-hand derivative means they are greater. On the right, a positive derivative means nearby values are greater than \(f(c)\), while a negative derivative means they are less.
| Left-hand behavior | Right-hand behavior | Conclusion at \(c\) |
|---|---|---|
| Increasing toward \(c\) | Decreasing away from \(c\) | Local maximum |
| Decreasing toward \(c\) | Increasing away from \(c\) | Local minimum |
| Increasing on both sides | Increasing on both sides | Neither, when the function rises through \(c\) |
| Decreasing on both sides | Decreasing on both sides | Neither, when the function falls through \(c\) |
“Increasing on the left” means that function values rise as \(x\) moves from left to right toward \(c\). It does not mean that values to the left are greater than \(f(c)\): they are lower. This distinction is essential when reading one-sided slopes.
Why Opposite Signs Give an Extremum
The difference quotient explains why nonzero one-sided derivatives classify the point. For \(h<0\), the denominator \(h\) is negative. For \(h>0\), it is positive. If the quotient has a nonzero limit, then sufficiently near \(c\) it has the same sign as that limit. The sign of \(h\) then determines whether \(f(c+h)-f(c)\) is positive or negative.
If \(f'_-(c)>0\), then for sufficiently small \(h<0\), the quotient is positive. Multiplying by \(h<0\) gives \(f(c+h)-f(c)<0\). If \(f'_-(c)<0\), then \(f(c+h)-f(c)>0\).
If \(f'_+(c)>0\), then for sufficiently small \(h>0\), the quotient is positive, so \(f(c+h)-f(c)>0\). If \(f'_+(c)<0\), then \(f(c+h)-f(c)<0\).
If nearby values on both sides are below \(f(c)\), there is a local maximum. If they are both above \(f(c)\), there is a local minimum. If one side is above and the other below, there is neither.
This argument is local: it concerns points sufficiently close to \(c\). It gives a complete classification whenever both one-sided derivatives exist and are nonzero. If a one-sided derivative is zero, the quotient’s limit alone does not determine the nearby values’ order; derivative signs on the neighboring intervals or direct comparison may be needed.
Worked Examples
Worked Example: A Corner That Is a Local Maximum
Classify the joining point \(x=1\) for
First check the value at the join. The left rule approaches \(1+2=3\), and the rule defining \(f(1)\) gives \(-2(1)+5=3\). Thus the function is continuous at \(1\), with \(f(1)=3\).
For \(x<1\), the derivative is \(1\), so \(f'_-(1)=1\). For \(x>1\), the derivative is \(-2\), so \(f'_+(1)=-2\). The function increases toward the join from the left and decreases from it on the right.
To check the value comparison directly, write \(x=1+h\). For \(h<0\), \(f(1+h)=(1+h)+2=3+h<3\). For \(h>0\), \(f(1+h)=-2(1+h)+5=3-2h<3\). Nearby values on both sides are below \(f(1)\), so \(x=1\) is a local maximum. The computed value \(f(1)=3\) agrees with both formulas at the join.
Worked Example: A Corner That Is a Local Minimum
Classify \(x=0\) for
At the join, the left rule approaches \(-0=0\), and the right rule gives \(g(0)=3(0)=0\). The function is continuous there. The left derivative is \(-1\), and the right derivative is \(3\).
For a direct check, if \(h<0\), then \(g(h)=-h>0=g(0)\). If \(h>0\), then \(g(h)=3h>0=g(0)\). Every sufficiently nearby value other than the value at zero is greater than \(g(0)\). Therefore \(x=0\) is a local minimum. The left side decreases toward the join; the right side increases away from it.
Worked Example: A Corner That Is Neither
Classify \(x=0\) for
Both pieces give \(p(0)=0\) by their limiting values, and the defined value is also \(0\). The left derivative is \(1\), while the right derivative is \(2\). Since the slopes are unequal, the ordinary derivative at zero does not exist.
If \(h<0\), then \(p(h)=h<0=p(0)\). If \(h>0\), then \(p(h)=2h>0=p(0)\). The nearby values lie below the function value on one side and above it on the other. Thus \(x=0\) is neither a local maximum nor a local minimum. The graph rises through the corner.
Worked Example: A Zero One-Sided Derivative Needs More Information
Classify \(x=0\) for
The function is continuous at zero, with \(q(0)=0\). On the left, \(q'(x)=-2x\), so \(q'_-(0)=0\). On the right, \(q'(x)=1\), so \(q'_+(0)=1\). The zero left-hand derivative does not by itself tell us whether nearby left values are above or below \(q(0)\).
For \(h<0\), \(q(h)=-h^2<0=q(0)\). For \(h>0\), \(q(h)=h>0=q(0)\). The values lie on opposite sides of \(q(0)\), so zero is neither a local maximum nor a local minimum. This example shows why a zero one-sided slope requires checking nearby behavior.
Continuity and the Actual Value at the Join
For a continuous piecewise function, the pieces meet at the point’s actual function value. The comparisons made using the one-sided slopes then refer to \(f(c)\), as in the examples. But if the function is discontinuous, the formulas on either side may approach a different value from \(f(c)\). In that situation, slopes alone do not settle the classification: compare nearby values directly with the value the function actually takes at \(c\).
For instance, a function can have increasing behavior to the left and decreasing behavior to the right, yet be assigned an unusually low value at the join. The graph’s nearby values would then be greater than \(f(c)\), giving a local minimum rather than the maximum suggested by the direction of the two pieces. Always check the joining value before classifying the point.
Common Mistake: Averaging the Two Slopes
A common error is to combine the two one-sided derivatives and treat their average as the derivative at the join. For \(r(x)=|x|\), for example, the left-hand derivative at zero is \(-1\) and the right-hand derivative is \(1\). A mistaken calculation might say
This conclusion is false. The ordinary derivative exists only if the two one-sided derivatives agree. Here, the left difference quotient is \(-1\) and the right difference quotient is \(1\), so their limits differ. Therefore \(r'(0)\) does not exist. The function nevertheless has a local minimum at zero, because \(|x|\ge 0=|0|\) for every \(x\).
Check Your Understanding
Use one-sided behavior and function-value comparisons to answer these questions.
- For a continuous piecewise function, what does a positive left-hand derivative say about nearby values to the left of the joining point?
- If the left-hand derivative is negative and the right-hand derivative is positive, and both are nonzero, how is the join classified?
- Why does a zero one-sided derivative require additional information?
- A piecewise function has left- and right-hand derivatives \(2\) and \(-3\) at a continuous join. What is the classification, and why?
- Why is it incorrect to average unequal one-sided derivatives to obtain the ordinary derivative?
