Free printable worksheets with answer keys. Pick a level, preview the questions, then open the sheet to swap or delete any single question.
Easy: Inflection point of a cubic
Where f″(x) > 0 the graph is concave up (it bends like a cup); where f″(x) < 0 it is concave down. An inflection point is where the concavity changes, so look for x-values where f″(x) = 0 and check that f″ changes sign there.
Use the second derivative to answer each question.
Tip: tick “Step-by-step answer key” in the generator to print each step for marking.
No. The concavity must actually change. f(x) = x⁴ has f″(0) = 0, but it is concave up on both sides, so (0, 0) isn't an inflection point.
Concave up means the slope is increasing; concave down means it is decreasing. An inflection point is where the slope stops increasing and starts decreasing, or the reverse.
Yes. Numbers are generated each time, and the answer key is calculated for that sheet. Swap any single question and its answer updates too.
Yes. In the generator, add more skills; each gets its own section on the sheet with its own instructions.
Yes. Generating and printing from the browser is free with no sign-up. PDF downloads need a free account.