Difficulty: Intermediate
Prerequisites: Module 1 (limits) and Module 2 (all differentiation rules). You must be able to find first and second derivatives fluently before tackling this module.
Big Picture
Module 3 is where derivatives become decision-making tools. The first derivative tells you whether a function is increasing or decreasing (direction). The second derivative tells you whether the curve is bending upward or downward (curvature/concavity). Together, they let you find the maximum and minimum values of cost, revenue, and profit functions, which is the core of optimisation in business. Elasticity of demand extends this further by measuring how sensitive quantity demanded is to price changes.
TL;DR
Set f'(x) = 0 to find critical points, use the second derivative to classify them as maxima or minima, and use the elasticity formula to determine whether raising or lowering prices will increase revenue.
Critical point
An x-value where f'(x) = 0 or f'(x) is undefined. These are the candidates for local maxima, local minima, or inflection points. Think of it as a point where the function pauses before changing direction.
Increasing function
A function is increasing on an interval when f'(x) > 0 throughout that interval. The output is getting larger as x moves to the right.
Decreasing function
A function is decreasing on an interval when f'(x) < 0 throughout that interval. The output is getting smaller as x moves to the right.
Concave up
A curve is concave up when f''(x) > 0. The curve opens upward like a bowl. A critical point where the curve is concave up is a local minimum.
Concave down
A curve is concave down when f''(x) < 0. The curve opens downward like an arch. A critical point where the curve is concave down is a local maximum.
Inflection point
A point where the concavity changes, from concave up to concave down or vice versa. Found where f''(x) = 0 and the sign of f'' actually changes.
Elasticity of demand (E)
A measure of how sensitive the quantity demanded is to a change in price. E = (p/q) × (dq/dp). When |E| > 1, demand is elastic (a price increase reduces revenue). When |E| < 1, demand is inelastic (a price increase raises revenue).
The first derivative f'(x) tells you the direction of the function:
f'(x) > 0: the function is increasing (going uphill left to right).
f'(x) < 0: the function is decreasing (going downhill left to right).
f'(x) = 0: the function has a horizontal tangent at that point. This is a critical point.
The second derivative f''(x) tells you the curvature:
f''(x) > 0: the curve is concave up (bowl shape). The rate of change is itself increasing.
f''(x) < 0: the curve is concave down (arch shape). The rate of change is itself decreasing.
f''(x) = 0: possible inflection point. Check whether f'' actually changes sign.
Second derivative test for classifying critical points
Find f'(x) and set it equal to zero. Solve for x to get the critical points.
Find f''(x).
Plug each critical point into f''(x):
If f''(x) > 0, the critical point is a local minimum.
If f''(x) < 0, the critical point is a local maximum.
If f''(x) = 0, the test is inconclusive; use the first derivative test instead.
The process for finding absolute or local extrema on a closed interval:
Find f'(x) and set it to zero. Solve for x.
Evaluate f(x) at each critical point and at the endpoints of the interval.
The largest value is the absolute maximum; the smallest is the absolute minimum.
For open-ended problems (no interval given), use the second derivative test to classify each critical point, or check the sign of f'(x) on either side of the critical point.
In business applications, this is how you find the production level that maximises profit or minimises cost. The critical point of a profit function P(x) where P'(x) = 0 and P''(x) < 0 is the output level that maximises profit.
Elasticity measures how responsive buyers are to price changes.
E = \frac{p}{q} \cdot \frac{dq}{dp}where p is price, q is quantity demanded, and dq/dp is the derivative of the demand function with respect to price.
|E| > 1: demand is elastic. A small price increase causes a proportionally larger drop in quantity. Raising the price decreases total revenue.
|E| < 1: demand is inelastic. A small price increase causes a proportionally smaller drop in quantity. Raising the price increases total revenue.
|E| = 1: unit elastic. Revenue is at its maximum at this price point.
The sign of E is typically negative (because demand falls as price rises), but the classification uses the absolute value.
Students assume every point where f'(x) = 0 is a max or min. It might be an inflection point or a saddle point instead. Always verify with the second derivative or a sign chart.
Concave up is often confused with increasing. A function can be concave up and decreasing (imagine the bottom of a bowl, left side). Concavity describes the bend, not the direction.
Students sometimes forget that elasticity uses the absolute value for classification. E is usually negative, but |E| > 1 still means elastic.
When finding extrema on a closed interval, students forget to check the endpoints. The absolute max or min might occur at an endpoint, not at a critical point.
⚠️ The second derivative test is a near-certain exam question. Practise the full workflow: find f'(x), set to zero, find f''(x), evaluate at critical points, classify.
⚠️ Elasticity of demand questions typically give you a demand function, ask you to compute E at a given price, and then ask whether the firm should raise or lower its price to increase revenue.
⚠️ Know the relationship between concavity and the second derivative cold. If f''(x) > 0, the curve is concave up (bowl). If f''(x) < 0, concave down (arch).
⚠️ Interval-based optimisation: do not skip the endpoints.
True or False: If f'(x) = 0, then x is a local maximum.
False. It is a critical point, but it could be a minimum or neither.
Fill in the blank: If f''(x) > 0 at a critical point, the point is a local ______.
Minimum.
True or False: Elastic demand (|E| > 1) means a price increase will raise total revenue.
False. Elastic demand means a price increase will decrease total revenue.
Fill in the blank: An inflection point is where the ______ changes sign.
Second derivative (f''(x)).
True or False: You only need to check critical points (not endpoints) to find the absolute max on a closed interval.
False. You must check both critical points and endpoints.
Q: Given f(x) = x³ - 12x + 5, find the critical points and classify each as a local max or min.
A: f'(x) = 3x² - 12. Set to zero: 3x² = 12, x² = 4, x = ±2. f''(x) = 6x. At x = 2: f''(2) = 12 > 0, so x = 2 is a local minimum. At x = -2: f''(-2) = -12 < 0, so x = -2 is a local maximum.
Q: A demand function is q = 500 - 10p. Find the elasticity of demand when p = 20.
A: dq/dp = -10. At p = 20, q = 500 - 200 = 300. E = (p/q)(dq/dp) = (20/300)(-10) = -2/3. |E| = 2/3 < 1, so demand is inelastic at this price. Raising the price would increase total revenue.
Q: On the interval [0, 5], find the absolute maximum of f(x) = -x² + 4x + 1.
A: f'(x) = -2x + 4. Set to zero: x = 2. Evaluate: f(0) = 1, f(2) = -4 + 8 + 1 = 5, f(5) = -25 + 20 + 1 = -4. The absolute maximum is 5 at x = 2.
Q: If f''(x) = 0 at x = 3, does that guarantee an inflection point?
A: No. You must also verify that f''(x) changes sign around x = 3. If f'' is positive on both sides (or negative on both sides), there is no inflection point.
Module 3 is the payoff for everything in Modules 1 and 2. The limit concept from Module 1 grounds the idea of instantaneous rate of change, and the differentiation rules from Module 2 are the tools you use to find f'(x) and f''(x). Optimisation (finding max/min) is the single most applied skill in business calculus: it determines optimal pricing, production levels, inventory quantities, and advertising spend.
Elasticity of demand connects to microeconomics. If you take an economics course later, the same formula reappears in discussions of market structure, taxation incidence, and welfare analysis.
critical points, first derivative test, second derivative test, concavity, concave up, concave down, inflection point, local maximum, local minimum, absolute maximum, absolute minimum, increasing, decreasing, elasticity of demand, elastic, inelastic, unit elastic, revenue optimisation, optimisation, closed interval method, MAC 2233, calculus for business, FIU, Professor Fox