Lecture 11 · Barnett ch. 4 · about 40 min

Second Derivative and Concavity

The derivative of the derivative: what concavity means, how to find inflection points, the second-derivative test as a shortcut for classifying extrema, and diminishing returns as a concavity story.

By the end you can

Compute a second derivative and interpret it as the rate of change of the rate of change
Determine concave-up and concave-down intervals from the sign of f double-prime
Find inflection points, including the point of diminishing returns
Apply the second-derivative test to classify critical points

The derivative of the derivative

Just as $f'(x)$ measures how $f$ is changing, $f''(x)$ measures how $f'(x)$ is changing: the rate at which the rate of change is itself changing.

Second derivativeFormula sheet →

f''(x) = \frac{d}{dx}\big[f'(x)\big]

$f''(x)$: second derivative of f, read 'f double prime'

Use it when you need to differentiate a function twice, such as to check concavity or apply the second-derivative test.

In economics, if $C(x)$ is total cost, $C'(x)$ is marginal cost, and $C''(x)$ tells you whether marginal cost itself is rising or falling — whether each additional unit is getting more or less expensive to produce than the one before.

Concavity

ConcavityRulesFormula sheet →

f''(x) \gt 0 \text{ on an interval} \;\Rightarrow\; f \text{ is concave up there} \qquad f''(x) \lt 0 \;\Rightarrow\; f \text{ is concave down there}

$concave up$: graph curves upward, like a cup
$concave down$: graph curves downward, like a frown

Use it when you need to describe the curvature of a graph, or confirm whether a critical point is a max or a min via the second-derivative test.

Inflection points

An inflection point is where concavity switches, from up to down or down to up. Just as critical numbers are candidates for extrema, points where $f''(x) = 0$ (or is undefined) are candidates for inflection points — and just as before, you must check that concavity actually changes.

Inflection pointDefinitionFormula sheet →

f''(c) = 0 \text{ (or undefined)}, \text{ and } f'' \text{ changes sign at } c

$c$: a candidate inflection point, whose sign change must still be confirmed, exactly as with the first-derivative test

Use it when a question asks for the point of diminishing returns, or any point where a curve changes from bending one way to the other.

The second-derivative test

Rather than building a full sign chart of $f'$ , you can often classify a critical point faster by checking the sign of $f''$ at that single point.

Second-derivative testKey ruleFormula sheet →

f'(c) = 0 \text{ and } f''(c) \gt 0 \;\Rightarrow\; \text{local min at } c \qquad f'(c)=0 \text{ and } f''(c) \lt 0 \;\Rightarrow\; \text{local max at } c

$if f''(c)$: 0, the test is inconclusive and the first-derivative test must be used instead

Use it when you already have f'' available (or it is easy to compute) and just need to classify a single critical point, rather than build a whole sign chart.

Exam practice

Summary and review

$f''(x)$ is the derivative of $f'(x)$ : the rate of change of the rate of change.
$f'' \gt 0$ : concave up. $f'' \lt 0$ : concave down.
Inflection point: $f''(c) = 0$ (or undefined) and concavity actually changes there.
The point of diminishing returns on a production function is an inflection point of output with respect to the input.
Second-derivative test: $f'(c)=0$ and $f''(c)\gt0$ gives a local min; $f'(c)=0$ and $f''(c)\lt0$ gives a local max; $f''(c)=0$ is inconclusive, fall back to the first-derivative test.