ECON5005 · Quantitative Tools for Economics
Curve Sketching & Concavity
Curve sketching and concavity is the Week 4 curve-analysis topic in ECON5005 Quantitative Tools for Economics at the University of Sydney. It shows you how to read a function's shape straight from the signs of its first and second derivatives — where it rises or falls, where it bends up (convex) or down (concave), and where it turns or inflects. These are the same tools the unit uses for optimisation, and curve analysis is examined in both the in-person mid-semester test and the final.
What this chapter covers
- 01Read slope from the first derivative: f′>0 increasing, f′<0 decreasing, f′=0 stationary
- 02Read curvature from the second derivative: f″>0 convex (bowl), f″<0 concave (dome)
- 03Locate stationary points by solving f′(x)=0
- 04Classify a stationary point with the second-derivative test (max, min, or inconclusive)
- 05Find inflection points where f″ changes sign — and why f″=0 alone is not enough
- 06Build a sign table for f′ and f″ across each interval
- 07Follow the five-step sketching recipe to assemble the shape
- 08Translate curvature into economics: rising marginal cost, diminishing marginal utility
Classify the stationary points and concavity of a cubic
- +1Differentiate: f′(x) = 3x² − 3 and f″(x) = 6x.
- +1Stationary points: set f′=0, so 3(x²−1)=0, giving x = −1 and x = 1.
- +1Classify with f″: f″(−1) = −6 < 0 → local maximum; f″(1) = 6 > 0 → local minimum.
- +1Heights: f(−1) = −1 + 3 = 2, so max at (−1, 2); f(1) = 1 − 3 = −2, so min at (1, −2).
- +1Concavity: f″(x) = 6x is negative for x < 0 (concave) and positive for x > 0 (convex).
- +1Inflection: f″=0 at x = 0 and f″ changes sign there, so (0, 0) is an inflection point.
Key terms
- Increasing function
- A function whose values rise as x rises; equivalently f′(x) > 0 on that interval.
- Decreasing function
- A function whose values fall as x rises; equivalently f′(x) < 0 on that interval.
- Stationary point
- A point where f′(x) = 0, so the tangent is horizontal; it may be a local maximum, a local minimum, or a horizontal inflection.
- Convex (concave up)
- A bowl-shaped stretch where f″(x) > 0; any chord joining two points lies on or above the curve.
- Concave (concave down)
- A dome-shaped stretch where f″(x) < 0; any chord joining two points lies on or below the curve.
- Inflection point
- A point where the second derivative changes sign, so the curve switches between convex and concave; f″=0 there is necessary but not sufficient.
- Second-derivative test
- At a stationary point, f″<0 signals a local maximum and f″>0 a local minimum; if f″=0 the test is inconclusive and you use a sign chart of f′.
Curve Sketching & Concavity FAQ
How do I tell a maximum from a minimum without plotting the whole curve?
Find the stationary points by solving f′(x)=0, then check the second derivative at each. If f″ is negative the curve is concave there (a dome), so it is a local maximum; if f″ is positive the curve is convex (a bowl), so it is a local minimum. If f″=0 the test is inconclusive, and you look at how the sign of f′ changes across the point.
Is every point where f″=0 an inflection point?
No. f″=0 only makes a point a candidate. An inflection requires the second derivative to actually change sign there. For example y=x⁴ has f″=12x², which is zero at x=0 but stays non-negative on both sides, so the curve remains convex and x=0 is not an inflection point. Always confirm the sign flip.
Can AI help me with curve sketching in ECON5005?
Yes, as a study aid. Sia, the AskSia AI tutor, can explain the method step by step — how to differentiate, solve f′=0, run the second-derivative test, and build a sign table — and can check your reasoning on practice problems so you understand each move. It does not sit your quizzes, mid-semester test, or final exam for you and cannot promise a grade; use it to learn the technique, then work fresh problems yourself under timed conditions.
Studying with AI? Sia — free AI economics tutor works through ECON5005 step by step.
Exam move
Treat curve analysis as a fixed procedure rather than a puzzle: differentiate twice, solve f′=0 for stationary points, classify each with the sign of f″, then solve f″=0 and confirm a genuine sign change for inflections. Practise writing out the full sign table for cubics and simple rational or exponential functions until the steps are automatic, and always attach y-values and economic meaning (rising marginal cost, diminishing marginal utility) to the shape. Because the topic is examined on both the 60-minute mid-semester test and the 2-hour final, rehearse showing your sign work cleanly at roughly 1.5 to 2 minutes per mark. Confirm the exact assessment weights and the open/closed-book and calculator policy on your Canvas site.
