AP Calculus AB Concavity and the Second Derivative Worksheet (PDF)
This concavity practice worksheet for AP Calculus AB covers concavity and the second derivative (Common Core AP CALC AB 5.7, AP CALC AB 5.8, AP CALC AB 5.9), with a free PDF to print.
These 21 problems practice concavity, inflection points, the Second Derivative Test and connecting the graphs of f, f prime and f double prime (AP Calculus AB, topics 5.7, 5.8, 5.9). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.
Later problems connect graphs of f and its derivatives and use the second derivative in situations such as sales growth and marginal cost. Work in pencil, show every step, and check the answer key only after a full attempt.
AP Calculus AB · Common Core State Standards: AP CALC AB 5.7, AP CALC AB 5.8, AP CALC AB 5.9
Concavity practice worksheet: exercises 1 to 10
Exercise 1 – Concavity of a cubic
Find the intervals of concavity and the inflection point of \(f(x)=x^{3}-6x^{2}+9x+1\).
Exercise 2 – Concavity of a logarithm
Exercise 3 – Exponential inflection
Exercise 4 – Always concave down
Exercise 5 – Sine on an interval
Exercise 6 – Inflection where f double prime is undefined
Show that \(f(x)=\sqrt[3]{x}\) has an inflection point at \(x=0\), even though \(f^{\prime\prime}(0)\) does not exist.
Exercise 7 – The bell curve
Exercise 8 – Zero but not an inflection
Exercise 9 – Second derivative test, cubic
Exercise 10 – Second derivative test, exponential
Use the Second Derivative Test to classify the critical numbers of \(f(x)=x^{2}e^{-x}\).
Exercises 11 and up
Exercise 11 – A test that fails
Exercise 12 – Concavity from the graph of f prime
The graph shows \(f^{\prime}(x)=x^{3}-3x\). On which intervals is \(f\) concave up? Where are the inflection points of \(f\)?
Exercise 13 – Conclusions from a sign table
Describe the concavity of \(f\) and the inflection points.
Exercise 14 – All features of a cubic
Exercise 15 – Curve sketch
Find the intercepts, relative extrema and inflection point of \(f(x)=x^{3}-6x^{2}+9x\).
Exercise 16 – Finding coefficients
Exercise 17 – Growth of sales
A new game’s total sales in thousands of dollars after \(t\) months is \(S(t)=-t^{3}+9t^{2}+20t\). When is the sales rate the greatest, and what is it?
Exercise 18 – Logistic population
A bird population is \(P(t)=\frac{1000}{1+9e^{-t/2}}\) after \(t\) years. At what time does the population grow fastest? What is the population then?
Exercise 19 – Marginal cost
Exercise 20 – Both tests together
Exercise 21 – Justifying inflection points
Key facts behind this concavity practice worksheet
- A positive f″ means concave up (slope increasing); a negative f″ means concave down (slope decreasing).
- An inflection point needs a change in the sign of f″, not just f″ = 0.
- Second Derivative Test: f′(c) = 0 and a positive f″(c) gives a relative minimum; a negative f″(c) gives a maximum.
- If f″(c) = 0 the test fails; use the First Derivative Test.
How to use this concavity practice worksheet
Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.
Download the concavity practice worksheet as a PDF with the button below and keep it with your notes.
Download the concavity practice worksheet as a PDF with the button below and keep it with your notes.
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