Reading Extrema from Derivative Graphs

Reading the Derivative Graph

A graph of \(y=f'(x)\) shows the rate of change of \(f\), not the values of \(f\) itself. Its position relative to the horizontal axis tells us whether \(f\) is increasing or decreasing: where the graph lies above the axis, \(f'(x)>0\); where it lies below, \(f'(x)<0\). Where it meets the axis, \(f'(x)=0\).

These signs reveal how \(f\) behaves near a point. If \(f'\) changes from positive to negative, the function changes from increasing to decreasing, giving a local maximum. If \(f'\) changes from negative to positive, the function changes from decreasing to increasing, giving a local minimum. A zero where the sign does not change is different: it does not, by itself, mark a local extremum.

Definitions and the Sign Test

A function \(f\) is increasing on an interval if \(x_1<x_2\) in that interval implies \(f(x_1)<f(x_2)\). It is decreasing if \(x_1<x_2\) implies \(f(x_1)>f(x_2)\). These are strict forms of monotonicity; the corresponding non-strict definitions use \(\leq\) and \(\geq\).

An interior point \(c\) is a local maximum if \(f(c)\geq f(x)\) for all \(x\) sufficiently close to \(c\) in the domain. It is a local minimum if \(f(c)\leq f(x)\) for all such \(x\). The inequalities are strict for nearby \(x\ne c\) when the extremum is strict.

Suppose \(f\) is continuous near an interior point \(c\), differentiable on each side of \(c\), and the derivative has a fixed strict sign on each side. The derivative sign test is:

The test describes behavior on intervals on either side of \(c\). A derivative graph may touch the axis at \(c\) without crossing it, so the last two rows are essential: a zero of \(f'\) need not be an extremum.

Why the Derivative Sign Determines Monotonicity

The connection between derivative signs and increasing or decreasing behavior follows from the Mean Value Theorem. In the form needed here, if \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then some \(d\in(a,b)\) satisfies

If \(f'(x)>0\) throughout an interval, take any two points \(a<b\) in it. The hypotheses of the Mean Value Theorem hold on \([a,b]\), and the theorem gives a point \(d\in(a,b)\) such that

because both \(f'(d)\) and \(b-a\) are positive. Therefore \(f(a)<f(b)\), so \(f\) is strictly increasing. If \(f'(x)<0\) throughout the interval, the same equation gives \(f(b)-f(a)<0\), so \(f\) is strictly decreasing.

Now suppose the derivative is positive just to the left of \(c\) and negative just to its right. On the left, \(f\) is increasing, so values immediately before \(c\) are smaller than \(f(c)\); on the right, \(f\) is decreasing, so values immediately after \(c\) are also smaller than \(f(c)\). Thus \(c\) is a strict local maximum. Reversing the signs gives a strict local minimum. This is why the sign-change test works.

Worked Examples

A Zero Is a Candidate, Not a Conclusion

When \(f\) is differentiable at an interior local extremum \(c\), its derivative must be zero. To see why, suppose first that \(c\) is a local maximum. For small positive \(h\), \(f(c+h)-f(c)\leq 0\), so the difference quotient \(\frac{f(c+h)-f(c)}{h}\leq 0\). For small negative \(h\), the numerator is still nonpositive but the denominator is negative, so the quotient is \(\geq 0\). Since the derivative exists, the left and right limits of these quotients agree. Their common value must be both nonpositive and nonnegative, hence \(f'(c)=0\). The argument for a local minimum reverses the inequalities and gives the same conclusion.

This fact makes a zero of \(f'\) a possible location for an interior extremum, but not a guarantee. For example, in the second worked example, \(f'(0)=0\) while the sign remains negative on both sides. The function keeps decreasing through that point.

Common Mistake: Calling Every Zero an Extremum

A tempting but incorrect rule is: “If \(f'(c)=0\), then \(f\) has a maximum or minimum at \(c\).” Consider \(f(x)=\frac{x^3}{3}\). Differentiating gives

Thus \(f'(0)=0\). But \(f'(x)=x^2>0\) for every \(x\ne 0\), both to the left and right of \(0\). The function is increasing on both sides, so \(0\) is not a local extremum. Directly, for any \(x<0\), \(\frac{x^3}{3}<0=f(0)\), while for any \(x>0\), \(\frac{x^3}{3}>0=f(0)\). Values occur both below and above \(f(0)\) arbitrarily close to \(0\).

Correction: Locate zeros of \(f'\), then compare its signs on either side of each zero. A zero with no sign change does not produce a local maximum or minimum under the sign test.

What a Derivative Graph Does and Does Not Show

A derivative graph gives sign and rate-of-change information. It can show increasing and decreasing intervals and, through sign changes, local extrema. It does not give the actual height of the function graph: knowing \(f'\) does not specify the vertical value of \(f\) without additional information such as a known value \(f(a)\).

Nor does identifying a local maximum or minimum automatically establish an absolute maximum or minimum. Absolute extrema depend on comparisons across the full domain, including relevant endpoints or behavior toward the edges of the domain. The derivative graph is a tool for reading local behavior; any global conclusion requires the appropriate domain and comparisons.

When reading a graph, first mark where it meets the axis, then inspect the sign on every interval between those points. At each zero, record the sign immediately before and after it. This avoids confusing a touch of the axis with a crossing and keeps the classification tied to the function’s actual change in direction.