3.1 Concave and convex functions of a single variable
Definitions
The twin notions of concavity and convexity are used widely in economic theory, and are also central to optimization theory. A function of a single variable is concave if every line segment joining two points on its graph does not lie above the graph at any point. Symmetrically, a function of a single variable is convex if every line segment joining two points on its graph does not lie below the graph at any point. These concepts are illustrated in the following figures.
Here is a precise definition.
 Definition

Let f be a function of a single variable defined on an interval. Then f is
 concave if every line segment joining two points on its graph is never above the graph
 convex if every line segment joining two points on its graph is never below the graph.
Denote the height of the line segment from (a, f(a)) to (b, f(b)) at the point x by h_{a,b}(x). Then according to the definition, the function f is concave if and only if for every pair of numbers a and b with x_{1} ≤ a ≤ x_{2} and x_{1} ≤ b ≤ x_{2} we have
f(x)  ≥  h_{a,b}(x) for all x with a ≤ x ≤ b. (*) 
h_{a,b}((1 − λ)a + λb)  =  (1 − λ)h_{a,b}(a) + λh_{a,b}(b) 
h_{a,b}((1 − λ)a + λb)  =  (1 − λ)f(a) + λf(b). 
f((1−λ)a + λb)  ≥  (1 − λ)f(a) + λf(b) for all λ with 0 ≤ λ ≤ 1. 
 Definition

Let f be a function of a single variable defined on the interval I. Then f is
 concave if for all a ∈ I, all b ∈ I, and all λ ∈ (0, 1) we have
f((1−λ)a + λb) ≥ (1 − λ)f(a) + λf(b)  convex if for all a ∈ I, all b ∈ I, and all λ ∈ (0, 1) we have
f((1−λ)a + λb) ≤ (1 − λ)f(a) + λf(b).
 concave if for all a ∈ I, all b ∈ I, and all λ ∈ (0, 1) we have
Note that a function may be both concave and convex. Let f be such a function. Then for all values of a and b we have
f((1−λ)a + λb)  ≥  (1 − λ)f(a) + λf(b) for all λ ∈ (0, 1) 
f((1−λ)a + λb)  ≤  (1 − λ)f(a) + λf(b) for all λ ∈ (0, 1). 
f((1−λ)a + λb)  =  (1 − λ)f(a) + λf(b) for all λ ∈ (0, 1). 
Economists often assume that a firm's production function is increasing and concave. Examples of such a function for a firm that uses a single input are shown in the next two figures. The fact that such a production function is increasing means that more input generates more output. The fact that it is concave means that the increase in output generated by each oneunit increase in the input does not increase as more input is used. In economic jargon, there are “nonincreasing returns” to the input, or, given that the firm uses a single input, “nonincreasing returns to scale”. In the example in the first of the following two figures, the increase in output generated by each oneunit increase in the input not only does not increase as more of the input is used, but in fact decreases, so that in economic jargon there are “diminishing returns”, not merely “nonincreasing returns”, to the input.
The notions of concavity and convexity are important in optimization theory because, as we shall see, a simple condition is sufficient (as well as necessary) for a maximizer of a differentiable concave function and for a minimizer of a differentiable convex function. (Precisely, every point at which the derivative of a concave differentiable function is zero is a maximizer of the function, and every point at which the derivative of a convex differentiable function is zero is a minimizer of the function.)
The next result shows that a nondecreasing concave transformation of a concave function is concave.
 Proposition
 Let U be a concave function of a single variable and g a nondecreasing and concave function of a single variable. Define the function f by f(x) = g(U(x)) for all x. Then f is concave.
 Proof

We need to show that f((1−λ)a + λb) ≥ (1−λ)f(a) + λf(b) for all values of a and b with a ≤ b and all λ ∈ (0, 1).
By the definition of f we have
f((1−λ)a + λb) = g(U((1−λ)a + λb)). U((1−λ)a + λb) ≥ (1 − λ)U(a) + λU(b). g(U((1−λ)a + λb)) ≥ g((1−λ)U(a) + λU(b)). g((1−λ)U(a) + λU(b)) ≥ (1−λ)g(U(a)) + λg(U(b)) = (1−λ)f(a) + λf(b).So f is concave.
Jensen's inequality: another characterization of concave and convex functions
If we let λ_{1} = 1 − λ and λ_{2} = λ in the earlier definition of a concave function and replace a by x_{1} and b by x_{2}, the definition becomes: f is concave on the interval I if for all x_{1} ∈ I, all x_{2} ∈ I, and all λ_{1} ≥ 0 and λ_{2} ≥ 0 with λ_{1} + λ_{2} = 1 we havef(λ_{1}x_{1} + λ_{2}x_{2})  ≥  λ_{1}f(x_{1}) + λ_{2}f(x_{2}). 
 Proposition (Jensen's inequality)

A function f of a single variable defined on the interval I is concave if and only if for all n ≥ 2
f(λ_{1}x_{1} + ... + λ_{n}x_{n}) ≥ λ_{1}f(x_{1}) + ... + λ_{n}f(x_{n})
i=1λ_{i} = 1.The function f of a single variable defined on the interval I is convex if and only if for all n ≥ 2
f(λ_{1}x_{1} + ... + λ_{n}x_{n}) ≤ λ_{1}f(x_{1}) + ... + λ_{n}f(x_{n})
i=1λ_{i} = 1.
 Source
 The result is a special case of a result for functions of many variables.
Differentiable functions
The following diagram of a differentiable concave function should convince you that the graph of such a function lies on or below every tangent to the function. In the figure, the red line is the graph of the function and the gray line is the tangent at the point x*, which has slope f'(x*).
The fact that the graph of the function lies below this tangent is equivalent to
The next result states this observation, and the similar one for convex functions, precisely. It is used to show the important result that for a concave differentiable function f every point x for which f'(x) = 0 is a global maximizer, and for a convex differentiable function every such point is a global minimizer.
 Proposition

The differentiable function f of a single variable defined on an open interval I is concave on I if and only if
f(x) − f(x*) ≤ f'(x*)(x − x*) for all x ∈ I and x* ∈ I f(x) − f(x*) ≥ f'(x*)(x − x*) for all x ∈ I and x* ∈ I.
 Proof

I first show that if f is concave on I then the first inequality in the result holds. From the definition of a concave function, for all x ∈ I, all x* ∈ I, and all λ ∈ (0, 1) we have
f((1 − λ)x* + λx) ≥ (1 − λ)f(x*) + λf(x) f((1 − λ)x* + λx) ≥ f(x*) + λ(f(x) − f(x*)) f(x) − f(x*) ≤ f((1 − λ)x* + λx) − f(x*) λ . g(λ) = f((1 − λ)x* + λx). f(x) − f(x*) ≤ (g(λ) − g(0))/λ. f(x) − f(x*) ≤ g'(0). g'(λ) = f'((1 − λ)x* + λx)(x − x*),sog'(0) = f'(x*)(x − x*).Substituting this expression for the righthand side of the inequality two lines above, we get the inequality in the result.I now show that if the first inequality in the result holds then f is concave. Let x* ∈ I and x ∈ I, and define x' = (1 − λ)x* + λx. Then x' ∈ I and by the inequality, which holds for all values of x and x* in I, we have both
f(x*) − f(x') ≤ f'(x')(x* − x') f(x) − f(x') ≤ f'(x')(x − x') (1 − λ)(f(x*) − f(x')) + λ(f(x) − f(x')) ≤ (1 − λ)f'(x')(x* − x')) + λf'(x')(x − x') (1 − λ)f(x*) + λf(x) − f(x') ≤ f'(x')((1 − λ)x* + λx − x'). (1 − λ)f(x*) + λf(x) ≤ f((1 − λ)x* + λx), Symmetric arguments apply for a convex function.
Twicedifferentiable functions
We often assume that the functions in economic models (e.g. a firm's production function, a consumer's utility function) are twicedifferentiable. We may determine the concavity or convexity of such a function by examining its second derivative: a function whose second derivative is nonpositive everywhere is concave, and a function whose second derivative is nonnegative everywhere is convex. Proposition
 A twicedifferentiable function f of a single variable defined on the interval I is
 Source
 The result is a special case of a result for functions of many variables. For a direct proof, see Rockafellar (1970), Theorem 4.4 (p. 26).
 Example
 Is x^{2} − 2x + 2 concave or convex on any interval? Its second derivative is 2 ≥ 0, so it is convex for all values of x.
 Example
 Is x^{3} − x^{2} concave or convex on any interval? Its second derivative is 6x − 2, so it is convex on the interval [1/3, ∞) and concave the interval (−∞, 1/3].
 Proposition
 Let U be a concave function of a single variable and g a nondecreasing and concave function of a single variable. Assume that U and g are twicedifferentiable. Define the function f by f(x) = g(U(x)) for all x. Then f is concave.
 Proof

We have f'(x) = g'(U(x))U'(x), so that
f"(x) = g"(U(x))·U'(x)·U'(x) + g'(U(x))U"(x).
 Definition

The point c is an inflection point of a twicedifferentiable function f of a single variable if f"(c) = 0 and for some values of a and b with a < c < b we have
 either f"(x) > 0 if a < x < c and f"(x) < 0 if c < x < b
 or f"(x) < 0 if a < x < c and f"(x) > 0 if c < x < b.
Note that some authors, including Sydsæter and Hammond (1995) (p. 308), give a slightly different definition, in which the conditions f"(x) > 0 and f"(x) < 0 are replaced by f"(x) ≥ 0 and f"(x) ≤ 0. According to this alternative definition, f" does not have to change sign at c. For example, for a linear function, every point satisfies the alternative definition.
Strict convexity and concavity
The inequalities in the definition of concave and convex functions are weak: such functions may have linear parts, as does the function in the following figure for x > a.
A concave function that has no linear parts is said to be strictly concave.
 Definition

The function f of a single variable defined on the interval I is
 strictly concave if for all a ∈ I, all b ∈ I with a ≠ b, and all λ ∈ (0,1) we have
f((1−λ)a + λb) > (1 − λ)f(a) + λf(b).  strictly convex if for all a ∈ I, all b ∈ I with a ≠ b, and all λ ∈ (0,1) we have
f((1−λ)a + λb) < (1 − λ)f(a) + λf(b).
 strictly concave if for all a ∈ I, all b ∈ I with a ≠ b, and all λ ∈ (0,1) we have