What is the difference between a local and a global maximum?

A local maximum is the highest point in its neighbourhood. The global maximum is the highest value over the whole interval, which may be a local maximum or an end point.

Does DSE M1 require curve sketching with asymptotes?

The M1 learning objective is to use differentiation for tangents, rates of change, maxima and minima. Curve sketching of rational functions with asymptotes is an M2 objective.

Which terms from other subjects can appear without definition?

Displacement, velocity and acceleration. Other terms from other disciplines should be defined in the question.

Applications of Differentiation (DSE M1): Maximum, Minimum and Rates of Change

To find a maximum or minimum, solve f′(x) = 0, classify each point with the first or second derivative, and for a global extreme on an interval also compare the values at the end points. A rate of change is a derivative with respect to time.

Solve Applications of Differentiation questions with AI

The tangent to y = x e⁻ˣ flattens at x = 1, where Example 1 finds the greatest value 1/e. Try it yourself →

Applications of Differentiation: Key Facts

Stationary pointf′(x) = 0

Second derivative testf″ < 0: maximum; f″ > 0: minimum

Global extreme on [a, b]Compare f(a), f(b) and the values at the stationary points

Velocity and accelerationv = ds/dt, a = dv/dt

Worked Examples — Applications of Differentiation

Example 1: A global maximum on an interval

Find the greatest value of f(x) = x e⁻ˣ for 0 ≤ x ≤ 3.

f′(x) = (1 − x)e⁻ˣ
f′(x) = 0 gives x = 1, and f″(1) = −e⁻¹ < 0, so it is a maximum
f(1) = 1/e ≈ 0.368
End points: f(0) = 0, f(3) = 3e⁻³ ≈ 0.149

Answer: The greatest value is 1/e, at x = 1.

Example 2: A rate of change

The population of a colony after t days is N = 500e⁰·²ᵗ. Find the rate of increase of the population when t = 5.

dN/dt = 500 × 0.2 × e⁰·²ᵗ = 100e⁰·²ᵗ
At t = 5: dN/dt = 100e¹
≈ 271.8

Answer: About 272 per day

💡 Exam Traps & Secrets

⚠️ Check the end pointsOn a closed interval the greatest or least value can occur at an end point where the derivative is not zero. A complete answer compares the stationary values with f(a) and f(b).

⚠️ Justify maximum or minimumSolving f′(x) = 0 only finds candidates. State the sign of f″, or the change of sign of f′, to earn the mark for the conclusion.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Global extrema)Find the maximum and minimum values of f(x) = x³ − 6x² + 9x + 1 for 0 ≤ x ≤ 5.

(Velocity and acceleration)A particle moves with displacement s = t³ − 6t² + 9t. Find when it is at rest and its acceleration then.

(Optimisation)The profit is P = 40x − x² − 100 (thousand dollars). Find the production level x that maximises profit.