Reading Extrema from Graphs and Derivative Graphs

Two Graphs, Two Ways to Read Extrema

A graph of a function shows the function’s output at each input. Near a local maximum, the graph rises toward a point and falls away from it; near a local minimum, it falls toward a point and rises away. A graph of the derivative shows something different: the slope of the function. Its signs tell us whether the function is increasing or decreasing, so a change in sign can reveal a local extremum even when the function’s graph is not shown.

These readings answer local questions. A point can be a local maximum without being the greatest function value on the entire domain, and a local minimum without being the least. The graph or derivative information must also be interpreted on the function’s domain: a point outside the domain cannot be a local extremum.

Reading a Function Graph Directly

To read extrema from a function graph, compare the height of a point with the heights at nearby inputs. A local maximum is a nearby peak; a local minimum is a nearby valley. The point’s coordinates are \((c,f(c))\): the input \(c\) tells where the extremum occurs, and the output \(f(c)\) tells its value.

Do not decide from height alone. The tallest point visible in a cropped graph is not necessarily a local maximum if the graph continues rising on both sides of that point. Conversely, a local peak need not be the highest point in the whole displayed domain. A graph’s scale, open circles, gaps, and endpoints also matter: an open circle is not a function value, while an included endpoint can be an extremum under a one-sided comparison with the domain.

Reading a Derivative Graph

For a differentiable function, \(f'(x)>0\) means that \(f\) is increasing on the relevant interval, while \(f'(x)<0\) means that \(f\) is decreasing there. On a derivative graph, these signs are read by checking whether the graph of \(f'\) lies above or below its horizontal axis. The vertical height of the derivative graph is a slope, not the value of \(f\).

At a point where \(f'\) changes from positive to negative, the function changes from increasing to decreasing, indicating a local maximum. A change from negative to positive indicates a local minimum. The location is the input at which the sign changes; it is not the derivative graph’s vertical coordinate.

The continuity and differentiability conditions matter when applying this rule. A derivative graph can be used only for inputs where the derivative is defined, and a point where \(f'\) is undefined requires separate care. The sign-change conclusion applies when the stated conditions hold; it is not a claim that every zero or undefined value of \(f'\) is an extremum.

Why the Sign Change Gives an Extremum

The reason is that a positive derivative means nearby function values increase as \(x\) increases, while a negative derivative means they decrease. The Mean Value Theorem makes the comparison precise. For a local maximum, take a nearby input \(x<c\). If \(f'(t)>0\) for \(x<t<c\), continuity on \([x,c]\) and differentiability on \((x,c)\) allow the theorem to give some \(\xi\in(x,c)\) such that

Thus \(f(x)<f(c)\) on the left. For a nearby input \(x>c\), if \(f'(t)<0\) for \(c<t<x\), the theorem gives some \(\eta\in(c,x)\) such that

Thus \(f(x)<f(c)\) on the right as well, proving a strict local maximum. For a change from negative to positive, the same comparisons reverse: nearby function values on both sides are greater than \(f(c)\), so \(c\) is a strict local minimum.

Worked Example: Classify Two Turning Points from a Derivative Graph

Let \(f(x)=\frac{x^3}{3}-\frac{7x^2}{2}+10x\). Its derivative graph crosses the horizontal axis at \(x=2\) and \(x=5\). Use the derivative signs to classify the corresponding points, then find their function values.

First differentiate and factor:

For \(x<2\), both factors are negative, so \(f'(x)>0\). For \(2<x<5\), the first factor is positive and the second negative, so \(f'(x)<0\). For \(x>5\), both factors are positive, so \(f'(x)>0\). Therefore, the sign changes from positive to negative at \(x=2\), and from negative to positive at \(x=5\). There is a local maximum at \(x=2\) and a local minimum at \(x=5\).

Evaluate the function at the two inputs:

Check: Differentiating the given function returns \(x^2-7x+10=(x-2)(x-5)\), so the sign analysis uses the correct derivative. The local maximum point is \((2,\frac{26}{3})\); the local minimum point is \((5,\frac{25}{6})\). These values are consistent with the function increasing, decreasing, and then increasing.

A Derivative Zero Without an Extremum

A derivative graph can meet the horizontal axis without crossing it. In that case, its value is zero at the contact point, but its sign may be the same on both sides. A zero derivative alone therefore does not establish a local maximum or minimum. The sign on both sides is the decisive information for this method.

Common Pitfall: Treating Every Axis Crossing as a Maximum

A crossing of the derivative graph’s horizontal axis is not automatically a maximum. The direction of the crossing matters. For example, if \(f'(x)\) changes from negative to positive, then \(f\) changes from decreasing to increasing, so the point is a local minimum, not a maximum.

Another common error is confusing the derivative graph’s height with the function’s height. If the derivative graph crosses its axis at \(x=2\), it tells us that the slope is zero there; it does not tell us \(f(2)=0\). To find the function’s extremum value, read it from the function graph or evaluate \(f(2)\) using the function itself.

Finally, a derivative graph identifies behavior only where its signs and the relevant function conditions are known. If a graph is incomplete, has an unclear scale, or omits information about whether a point belongs to the domain, do not infer more than it shows.