Concavity and Convexity of Real Functions#

Concavity#

Definition: Concave

A real function \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) is concave on an interval \(I \subseteq \mathcal{D}\) if

\[ f(\lambda c + (1-\lambda)d) \ge \lambda f(c) + (1-\lambda)f(d) \]

for all \(\lambda \in [0; 1]\) and all \(c, d \in I\).

We say that \(f\) is strictly concave if we can replace \(\ge\) with \(\gt\).

Geometrically, the graph of a concave function always lies above any secant line:

Concave Function Graph

Theorem: Concavity via Differentiation

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function.

If \(f\) is continuous on a closed interval \([a;b]\) and twice continuously differentiable on the open interval \((a;b)\), then it is concave on \([a;b]\) if and only if

\[ f''(x) \le 0 \qquad \forall x \in (a;b) \]

and it is strictly concave if

\[ f''(x) \lt 0 \qquad \forall x \in (a;b). \]

Proof

TODO

Convexity#

Definition: Convexity

A real function \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) is convex on a closed interval \([a;b] \subseteq \mathcal{D}\) if

\[ f(\lambda c + (1-\lambda)d) \le \lambda f(c) + (1-\lambda)f(d) \]

for all \(\lambda \in [0;1]\) and all \(c, d \in I\).

We say that \(f\) is strictly convex if we can replace \(\le\) with \(\lt\).

Geometrically, the graph of a convex function always lies below any secant line:

Convex Function Graph

Theorem: Convexity via Differentiation

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function.

If \(f\) is continuous on a closed interval \([a;b]\) and twice continuously differentiable on the open interval \((a;b)\), then it is convex on \([a;b]\) if and only if

\[ f''(x) \ge 0 \qquad x \in (a;b) \]

and it is strictly convex if

\[ f''(x) \gt 0 \qquad x \in (a;b) \]

Proof

TODO

Example: \(x^2\)

The function \(f: \mathbb{R} \to \mathbb{R}\) defined as

\[ f(x) = x^2 \]

is strictly convex on all closed intervals \([a;b]\) because

\[ f''(x) = 2 \]

and so

\[ f''(x) \gt 0 \]

for all \(x \in (a;b)\).

Example: \(\mathrm{e}^x\)

The real exponential function is strictly convex on all closed intervals \([a;b]\) because

\[ (\mathrm{e}^x)'' = \mathrm{e}^x \]

and \(\mathrm{e}^x \gt 0\) for all \(x \in \mathbb{R}\).

Inflection Points#

Definition: Inflection Point

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) and let \(p \in \mathcal{D}\).

We say that \((p, f(p))\) is an inflection point of \(f\) if there exists some \(\varepsilon \gt 0\) such that \(f\) is convex on \((p-\varepsilon; p)\) and concave on \((p; p + \varepsilon)\) or vice versa.

Theorem: Inflection Points and Second Derivative

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function and let \(p \in \mathcal{D}\).

If \(f\) has an inflection point at \(p\), is differentiable on some open neighborhood of \(p\) and is twice differentiable at \(p\), then \(f''(p) = 0\).

Proof

TODO

Theorem: Second-Order Derivative Test for Inflection Points

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function and let \(p \in \mathcal{D}\).

If \(f\) is twice continuously differentiable on some open neighborhood of \(p\) with \(f''(p) = 0\) and there exists some \(\varepsilon \gt 0\) such that \(f''(x_1) \cdot f''(x_2) \lt 0\) for all \(x_1 \in (p - \varepsilon; p)\) and all \(x_2 \in (p; p + \varepsilon)\), then \(f\) has an inflection point at \(p\).

Proof

TODO

Theorem: Higher-Order Derivative Test for Inflection Points

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function, let \(p \in \mathcal{D}\).

Suppose \(f\) is \((n-1)\)-times differentiable \((n \ge 3)\) on some open neighborhood of \(p\) and is \(n\)-times differentiable at \(p\).

If \(f^{(k)}(p) = 0\) for all \(2 \leq k < n\) and \(f^{(n)}(p) \neq 0\), and if \(n\) is odd, then \(f\) has an inflection point at \(p\).

Proof

TODO