A function is defined as strictly convex if its second derivative is strictly greater than zero for all values of its input variable, x. This condition is mathematically expressed as $f''(x) > 0$. This is a more stringent condition than that for a standard convex function and implies that the function's slope strictly increases as the input variable increases.

Strictly Convex Function

A convex decreasing function is characterized by a slope that, while always negative, increases (becomes less negative) as the input variable `x` increases. This causes the function's curve to become progressively flatter. This characteristic corresponds to a positive second derivative ($f''(x) > 0$). In economics, this is the standard mathematical representation for the shape of most indifference curves.

Convex Decreasing Function

If a firm's total cost function is convex, it means that the marginal cost—represented by the slope of the cost curve—is an increasing function of the quantity of output produced.

Convexity of the Cost Curve Implies Increasing Marginal Cost

A function, $f(x)$, is defined as convex if its second derivative is non-negative ($f''(x) \ge 0$), which means its slope ($f'(x)$) is non-decreasing. A simple visual test for convexity is to draw a straight line segment between any two points on the function's graph; if the curve always lies on or below this connecting line, the function is convex.

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A derivative of a derivative is known as second derivative. Second derivative is often used to measure curvature.

Second Derivative

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

CORE Econ - The Economy 2.0: Microeconomics

The second derivative in a speciﬁc direction represented by a unit vector $d$ is given by $d^THd$.

Second Derivative in a Specific Direction

The second derivative can be used to determine whether a critical point is a local maximum, a local minimum, or a saddle point. We call such a test second derivative test.

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