Florida B.E.S.T. MA.912.C.3.5

MathGrades 9–12Apply derivatives to solve problems.

The Standard

Determine the concavity and points of inflection of a function using its second derivative.

Florida B.E.S.T. Standards for Mathematics

Teacher's field guide

What Students Need to Do: Students use the sign of the second derivative to describe how a graph bends on different intervals. They locate possible change points, test both sides, and give the coordinates where the bending direction changes.
What Mastery Looks Like: Students correctly find or use the second derivative, split the domain at possible change points, and test each interval. They report intervals with proper notation and identify only points where concavity changes.
Common Misconceptions: Students may think every value where the second derivative equals zero is an inflection point. They also confuse increasing with concave up and forget to check for a sign change.
How to Assess It: Give the exit ticket: “For f(x) = x⁴ − 2x³, make a sign chart for f″, state the concavity intervals, and find each inflection point.”

Lesson moves

Give groups cards showing functions, second derivatives, sign charts, and graphs, then have them match each complete set.
Ask students to explain why x = 0 is not an inflection point for f(x) = x⁴, even though f″(0) = 0.
Run a sign-chart relay where teams find candidates, test intervals, label concavity, and verify inflection points for four functions.
Analyze a water-tank height graph and connect concavity to whether the filling rate is increasing or decreasing.

Free download

$Preview of the MA.912.C.3.5 printable worksheet$

A ready-to-print activity worksheet aligned to MA.912.C.3.5, with an answer key for the teacher on its own page. No account needed.

PDF, US Letter, prints cleanly in black and white.

Keep exploring

Grade, subject, topic, and the complete standard are prefilled. Create one free, no account needed.