Do not classify from a zero derivative alone

Two horizontal tangents

The top graph shows \(f(x) = 4x^3 - 3x^4\) with its tangent at the orange point, and the bottom graph shows the derivative \(f'\).

The window is the stretch between the two faint lines around the point. Inside it, the shading is green where \(f\) is above the orange dashed line at the point's height and red where it is below.

Move the point to \(x = 1\). The tangent is horizontal and the window is all red, so \(f(1) = 1\) is a local maximum.

Move the point to \(x = 0\). The tangent is also horizontal, but the window holds both colors, so \(f(0)\) is neither a local maximum nor a local minimum.

The same slope, two different answers

The numbers are gone. The S-curve has a horizontal tangent at the orange point, and the window shows both colors, however narrow you make it.

Bend the curve into a bowl. The tangent is still horizontal, but the window shows only green, so the point is a local minimum.

A horizontal tangent only marks a point to test. The values on both sides decide whether the point is a maximum, a minimum or neither.

Notation

1. A critical point is an input where \(f'\) is 0 or does not exist. It is a candidate for a local extremum: a local maximum or minimum can be there but does not have to be.

2. If \(f\) has a local extremum at \(c\), and \(f'(c)\) exists, then \(f'(c) = 0\). On the bowl, the bottom is a local minimum and the tangent there is horizontal.

   \[\begin{aligned} &\text{local extremum at } c \\ &\text{and } f'(c) \text{ exists} \\ &\Rightarrow \cy{f'(c) = 0} \end{aligned}\]

3. The reverse is false. On the S-curve, the tangent is horizontal at the point, but the window holds both colors, so there is no local extremum.

   \[f'(c) = 0 \ \not\Rightarrow \ \text{extremum at } c\]

4. To decide, compare \(f(c)\) with the values on both sides. If both are lower, \(f(c)\) is a local maximum. If both are higher, it is a local minimum. If one is lower and one is higher, it is neither.

When it fails

The button sharpens the bottom of the bowl into a corner. The point is still the lowest nearby, so it is a local minimum.

The two orange half-lines have different slopes, so the curve has no tangent at the corner and \(f'(c)\) does not exist. The rule \(f'(c) = 0\) needs \(f'(c)\) to exist, so an extremum can also sit where \(f'\) does not.

Example 1: two flat points

Decide whether each critical point of \(p(x) = x^4 - 4x^3\) is a local maximum, a local minimum or neither.

1. Differentiate and factor. The derivative is 0 at \(x = 0\) and \(x = 3\), so these are the critical points.

   \[p'(x) = 4x^2(x - 3)\]

2. Compare the values near \(x = 0\). The left value is above \(p(0)\) and the right value is below it, so \(x = 0\) is neither.

   \[\begin{aligned} p(-0.5) &= \tfrac{9}{16} \\ p(0) &= 0 \\ p(0.5) &= -\tfrac{7}{16} \end{aligned}\]

3. Both values near \(x = 3\) are above \(p(3) = -27\), so the window is all green and \(x = 3\) is a local minimum.

   \[\begin{aligned} p(2.5) &= -\tfrac{375}{16} \\ p(3) &= -27 \\ p(3.5) &= -\tfrac{343}{16} \end{aligned}\]

Example 2: a corner

Find the local extrema of \(g(x) = 2|x - 3| - 1\).

1. Write \(g\) in two pieces. For \(x < 3\), \(g(x) = 5 - 2x\). For \(x > 3\), \(g(x) = 2x - 7\).

   \[\begin{aligned} g'(x) &= -2 \quad (x < 3) \\ g'(x) &= 2 \quad (x > 3) \end{aligned}\]

2. A tempting wrong move: The slope is \(-2\) or 2, never 0, so \(g\) has no critical point and no local extremum.

   The window at \(x = 3\) is all green, because the V is higher on both sides of its corner. A local minimum can occur where the derivative does not exist.

3. The two slopes at \(x = 3\) differ, so \(g'(3)\) does not exist and \(x = 3\) is a critical point. Both sides are higher than \(g(3) = -1\), so it is a local minimum.

   \[\begin{aligned} g(2.9) &= -0.8 \\ g(3) &= -1 \\ g(3.1) &= -0.8 \end{aligned}\]

Practice

Decide whether \(w(x) = 2x + |x - 1|\) has a local extremum at \(x = 1\).

Move the point to \(x = 1\) and make the window small. Find the color on each side of the point.

Questions

The curve has a smooth peak at A and a horizontal tangent at B, where it falls on both sides. It has a pointed valley at C and a corner at D, where it rises on both sides.

1. At which labeled point is the tangent horizontal but there is no local extremum?

   • B
   • A
   • C
   Answer

   B

2. Which labeled point is a local minimum where \(f'\) does not exist?

   • C
   • D
   • A
   Answer

   C

3. For \(f(x) = (x - 1)^3(x + 2)\), the derivative is \(f'(x) = (x - 1)^2(4x + 5)\). Which statement is correct?

   • \(f\) has a local minimum at \(x = -\tfrac54\) and no local extremum at \(x = 1\).
   • \(f\) has a local minimum at both \(x = -\tfrac54\) and \(x = 1\).
   • \(f\) has a local minimum at \(x = 1\) only.
   • \(f\) has no local extremum.
   Answer

   \(f\) has a local minimum at \(x = -\tfrac54\) and no local extremum at \(x = 1\).

Summary

You treat a zero or missing derivative as a candidate, and you compare the values on both sides to decide.

This works because a curve can cross the height of a flat point, and a corner can be the lowest point nearby.

You use it before you name a maximum or a minimum at a critical point.

This is one of five skills in topic 5.2 of Which & Why: AP Calculus.