Maxima and Minima: Finding Extremes the Simple Way

Maxima and minima are the highest and lowest values a function attains — the peaks and valleys of its graph. Finding them is one of calculus's most practical jobs: the maximum profit, the minimum cost, the strongest beam, the fastest route. The derivative locates them because at a smooth peak or valley, the tangent is flat, so f'(x) = 0.

This matters because optimization is everywhere. Businesses maximize revenue, engineers minimize material, athletes minimize time. The theory here — critical points, then testing — is the engine behind every "find the best" problem in the chapters that follow.

Key Concepts

• Local maximum/minimum: a point higher (or lower) than all nearby points on the graph.
• Absolute maximum/minimum: the single highest (or lowest) value on the entire domain or interval being considered.
• Critical point: a point where f'(x) = 0 or f'(x) is undefined. Maxima and minima can only occur at critical points or at interval endpoints.
• Endpoints matter: on a closed interval [a, b], the absolute extremes may sit at a or b even though the derivative is not zero there.

How It Works

1. Step 1: Find all critical points by solving f'(x) = 0 and noting where f' is undefined.
2. Step 2: For absolute extrema on [a, b], evaluate f at every critical point inside the interval AND at both endpoints.
3. Step 3: Compare the values: the largest is the absolute maximum, the smallest the absolute minimum.
4. Step 4: For local extrema, use the first or second derivative test at each critical point.

Worked Example

Find the absolute maximum and minimum of f(x) = x³ − 3x on the interval [−2, 2].

1. Step 1: f'(x) = 3x² − 3 = 3(x² − 1). Setting to zero gives critical points x = 1 and x = −1.
2. Step 2: Evaluate f at the critical points and endpoints: f(−2) = −8 + 6 = −2; f(−1) = −1 + 3 = 2; f(1) = 1 − 3 = −2; f(2) = 8 − 6 = 2.
3. Step 3: The largest value is 2 (at x = −1 and x = 2) and the smallest is −2 (at x = −2 and x = 1).
4. Step 4: So the absolute maximum is 2 and the absolute minimum is −2. Note the maximum occurs both at a critical point and at an endpoint — checking endpoints was essential.

Common Misconceptions

• "Every critical point is a max or min." False — y = x³ has a critical point at x = 0 that is neither; the function just flattens momentarily and keeps rising.
• "Endpoints can be ignored." On a closed interval they often carry the absolute extreme, as the example shows.
• "A function always has a maximum." On an open interval like (0, 1), f(x) = x has no maximum — it approaches 1 but never reaches it.

Real-World Applications

• Business: finding the production level that maximizes profit from a profit function.
• Engineering: minimizing the material (surface area) needed for a container of fixed volume.
• Physics: the path of least time (a light ray refracting) is found by minimizing a travel-time function — this is Fermat's principle.
• Logistics: minimizing total delivery cost as a function of warehouse location.

To recap: solve f'(x) = 0 for critical points, evaluate the function at all critical points plus any endpoints, and compare. The biggest value is the maximum, the smallest the minimum.