Concavity from a Graph of f'

Reading Concavity from the Direction of \(f'\)

A graph of \(f'\) carries information about the shape of \(f\). The key is to follow whether the graph of \(f'\) rises, falls, or stays level as \(x\) increases. Rising \(f'\) means the slopes of \(f\) are getting larger, so \(f\) is concave up. Falling \(f'\) means those slopes are getting smaller, so \(f\) is concave down.

This is different from asking whether \(f'\) is above or below the horizontal axis. The height of \(f'\) tells us the sign of the slope of \(f\). Its direction of change tells us about concavity. Keeping those two readings separate is essential when interpreting a graph.

What “Rising” and “Falling” Mean

As we move from left to right on the graph, \(f'\) is increasing when its values do not decrease, and it is decreasing when its values do not increase. A graph that rises strictly has increasing values; one that falls strictly has decreasing values. A horizontal section is constant, so it meets both non-strict descriptions.

The concavity rule follows the monotonicity criterion from the earlier material on concavity and monotonicity of the first derivative. In the usual concavity convention, non-strict monotonicity matters: linear pieces count as both concave up and concave down. If a question asks for strict concavity, a linear interval is neither strictly concave up nor strictly concave down.

A graph of \(f'\) may have a corner where its direction changes. That does not prevent us from reading whether \(f'\) rises or falls on either side. We do not need \(f'\) itself to have a derivative at every point in order to use its monotonicity to infer concavity.

Why the Rule Works

Here is a justification using chord slopes. Suppose \(f'\) is nondecreasing on an interval and choose any \(a<b<c\) in that interval. Assume \(f\) is continuous on the relevant closed intervals and differentiable on their interiors. By the Mean Value Theorem, there are points \(u\in(a,b)\) and \(v\in(b,c)\) such that

Since \(u<v\) and \(f'\) is nondecreasing, \(f'(u)\leq f'(v)\). Therefore, the slope of the chord from \(a\) to \(b\) is no greater than the slope of the chord from \(b\) to \(c\). This is the chord-slope condition for \(f\) to be concave up. If \(f'\) is nonincreasing, the inequalities reverse, giving concavity down.

If \(f'\) is constant on an interval, say \(f'(x)=m\), the Mean Value Theorem gives \(f(b)-f(a)=m(b-a)\) for any two points \(a<b\) there. Thus \(f(x)-mx\) is constant on that interval, so \(f\) is a straight line there. Its chord slopes are equal, which is why the interval meets both non-strict concavity descriptions.

Worked Example: Three Straight Sections of \(f'\)

Turning Points of \(f'\)

A local maximum or minimum in the graph of \(f'\) often marks a transition between concavity directions. If \(f'\) rises on one side of a turning point and falls on the other, \(f\) changes from concave up to concave down. If \(f'\) falls and then rises, the direction of concavity changes the other way. These observations identify where concavity changes; they do not say that every turning point must be an isolated point or that every flat stretch counts as a change.

Flat Sections and Their Neighbours

A horizontal section of the graph of \(f'\) must not be mistaken for a gap in the concavity information. It says that \(f\) has constant slope on that interval, hence is linear there. But how the flat section joins the sections around it also matters. For instance, a rising section followed by a flat one leaves \(f'\) nondecreasing across the join, so \(f\) remains concave up across that combined interval.

A Common Mistake: Reading Height Instead of Direction

It is tempting to use whether the graph of \(f'\) lies above or below the \(x\)-axis to decide concavity. That reads the wrong feature. Consider \(f'(x)=3-x\) on \((0,1)\). Its graph is above the axis throughout, but it falls as \(x\) increases.

A mistaken conclusion would be: “\(f'\) is positive, so \(f\) is concave up.” Positivity means that \(f\) is increasing, not that it is concave up. The displayed values show that \(f'\) is decreasing, so \(f\) is concave down. The correction is to track the vertical change in the graph of \(f'\) as \(x\) moves from left to right, rather than just its position relative to the axis.

Reading a Graph Systematically

When given a graph of \(f'\), divide its domain into intervals on which it consistently rises, falls, or stays level. Then translate each direction into a concavity statement for \(f\). At a turning point, inspect the intervals on both sides instead of assigning concavity from the point alone. At a flat section, state that \(f\) is linear there; if the section joins a rising or falling segment, check whether the monotonicity of \(f'\) continues across the join.

Use only the intervals represented by the graph. If the graph has a break or the domain has a gap, conclusions from one side do not automatically extend across it. Finally, keep two readings distinct: the sign of \(f'\) describes whether \(f\) increases or decreases, while the trend of \(f'\) describes the concavity of \(f\).