Extrema and the Study of Functions

The Problem of Extrema

Finding a maximum or a minimum is one of the most practical mathematical problems. A company wants to maximize profit. An engineer wants to minimize energy consumption. A physicist seeks the configuration with minimal energy.

Mathematical analysis provides a systematic toolkit.

Necessary Conditions for Extrema

Fermat's Theorem: If $f$ is differentiable at the point $x_0$ and a local extremum is achieved at that point, then $f'(x_0) = 0$.

Points where $f' = 0$ or $f'$ does not exist are called critical points. Extrema can only occur at critical points. However, not every critical point is an extremum: $f(x) = x^3$, $f'(0) = 0$, but $x = 0$ is not an extremum (it is an inflection point).

Sufficient Conditions

First Derivative Test: If $f'$ changes sign from $+$ to $-$ as $x_0$ is passed through, then $x_0$ is a maximum. From $-$ to $+$ — a minimum. If the sign does not change — not an extremum.

Second Derivative Test: If $f'(x_0) = 0$ and $f''(x_0) > 0$, then $x_0$ is a minimum. If $f''(x_0) < 0$ — a maximum. If $f''(x_0) = 0$ — indeterminate, another test is needed.

Convexity and Inflection Points

A function is convex downward on an interval if $f'' > 0$. Convex upward — $f'' < 0$.

An inflection point is a point where convexity changes. At such a point $f'' = 0$ (necessary but not sufficient).

Scheme for Studying a Function

1. Domain of definition
2. Even/odd, periodicity
3. Zeros, sign of the function
4. Asymptotes (vertical, horizontal, oblique)
5. Monotonicity: $f' > 0$ — increasing, $f' < 0$ — decreasing
6. Extrema
7. Convexity, inflection points
8. Graph

Oblique asymptote: $y = kx + b$, where $k = \lim_{x \to \infty} f(x)/x$, $b = \lim_{x \to \infty} (f(x) - kx)$.

Global Extrema

On a closed interval $[a, b]$ the maximum and minimum are attained (Weierstrass theorem). They must be sought among:

• critical points inside the interval
• values at the endpoints $a$ and $b$

For unbounded regions, the situation is more complicated.

Practical Problems

Maximum Area Problem. From wire of length $L$ to make a rectangle of maximum area. For $2(a + b) = L$ the maximum $S = ab$ is achieved at $a = b = L/4$ (a square).

Minimum Cost Problem. Such problems arise in economics: minimizing costs under constraints.

Convexity and Economic Applications

A function is called convex on an interval if $f''(x) \ge 0$. Economic meaning: decreasing marginal productivity of capital — convexity of the production function. When the production function is concave ($f'' < 0$), marginal product decreases: each additional unit of capital gives less increase. This is the foundation of the law of diminishing returns.

Jensen's Inequality: For a convex function $f$: $f(\mathbb{E}[X]) \le \mathbb{E}[f(X)]$. Here, $\mathbb{E}$ is mathematical expectation. This inequality underlies many economic theorems: a risk-neutral agent values a lottery higher than a pessimist (with a concave utility function).

The Lagrange Multipliers Method (First Introduction)

When one minimizes a function subject to a constraint, "blindly" finding critical points does not work — the constraint must be taken into account. The method of Lagrange multipliers solves this systematically: add a term $\lambda \cdot$ (constraint) to the objective function. The optimum is achieved where the gradient of the objective function is parallel to the gradient of the constraint.

Economic Example: Maximize utility $U(x, y) = \sqrt{x} + \sqrt{y}$ subject to the budget constraint $px + qy = I$. The solution yields $x^* = I/(2p)$, $y^* = I/(2q)$ — the consumer spends half the income on each good (in this case). The Lagrange multiplier $\lambda = 1/(2\sqrt{pI})$ — this is the marginal utility of income: how the maximum utility changes when the budget increases by one unit.

Asymptotes and Behavior at Infinity

For rational functions $f(x) = P(x)/Q(x)$, the behavior as $x \to \infty$ is determined by the degrees of numerator and denominator. If $\deg P = \deg Q$: horizontal asymptote $y = a_n/b_n$. If $\deg P = \deg Q + 1$: oblique asymptote. If $\deg P > \deg Q + 1$: $f \to \infty$ (divergence).

Convexity and Second Derivative: Economic Applications

The sign of the second derivative $f''(x)$ determines convexity: $f'' > 0$ — convex downward (a “cup”-shaped curve, cost function with increasing marginal cost), $f'' < 0$ — convex upward (a “bell”-shaped curve, utility function with diminishing marginal utility). Inflection points — where $f''(x) = 0$ and the sign changes — these are moments of acceleration or deceleration in growth. In data analysis the inflection point on an S-shaped curve (logistic function) is the moment when the spread rate of a product/disease/information is maximal. The second derivative is the standard tool in behavioral economics for assessing the degree of risk aversion: an agent with a concave utility function ($u'' < 0$) is risk-averse.

Question for thought: The function $f(x) = x + \sin x$ does not have a horizontal asymptote, but also does not go to infinity “monotonically.” How to describe its behavior as $x \to \infty$?

Second Order Conditions in Optimization Problems

To distinguish minimum, maximum, and inflection, one uses second order conditions. If $f'(x_0) = 0$:

• for $f''(x_0) > 0$ — local minimum;
• $f''(x_0) < 0$ — local maximum;
• $f''(x_0) = 0$ — indeterminate, analyze the sign of $f'$.

In the multivariate case ($f: \mathbb{R}^n \to \mathbb{R}$) instead of $f''$ one uses the Hessian matrix $H$: if $H$ is positive definite at the critical point — it's a minimum; negative definite — a maximum; indefinite — a saddle point. In convex optimization and machine learning (SVM, logistic regression), the loss function is convex — its Hessian is positive semi-definite, which guarantees the absence of local minima other than the global one.