Concavity and Curve Sketching

Learn second-derivative signs, inflection points.

Lesson 3 of 32 parts Practice Flashcards

1/2

Second-Derivative Signs

Objective: Explain second-derivative signs, apply the stated method, and check its conditions.

Where f″>0, the first derivative increases and the graph is concave up; where f″<0, it is concave down. Concavity is different from rising or falling: a curve can decrease while bending upward. A second-derivative sign must be interpreted on an interval. Concavity and monotonicity describe different features.

Formulas

f″>0: concave up; f″<0: concave down

Worked examples

Try itFor f(x)=6x³−7x, evaluate f″ at x=7.Show answer

Step 1: f′(x)=18x²−7. Step 2: f″(x)=36x. Step 3: f″(7)=252>0, so it is concave up there.

Swipe sideways to move between sections

Second-Derivative Signs

Objective: Explain second-derivative signs, apply the stated method, and check its conditions.

Formulas

f″>0: concave up; f″<0: concave down

Worked examples

Try itFor f(x)=6x³−7x, evaluate f″ at x=7.Show answer

Step 1: f′(x)=18x²−7. Step 2: f″(x)=36x. Step 3: f″(7)=252>0, so it is concave up there.

Inflection Points

Objective: Explain inflection points, apply the stated method, and check its conditions.

An inflection point is a continuous graph point where concavity changes. A zero or undefined second derivative is a candidate, not a guarantee. Check opposite-side concavity and verify that the point belongs to the graph before identifying an inflection. f″=0 alone does not establish an inflection point.

Formulas

Inflection requires a change of concavity

Worked examples

Try itFor f(x)=(x−7)³+7, find the x-coordinate of its inflection point.Show answer

Step 1: f″(x)=6(x−7). Step 2: The second derivative changes sign at x=7. Step 3: The inflection x-coordinate is 7.

You finished the reading!

Lock it in with a few practice questions. This is the last lesson in the chapter.

Practice

Previous lessonExtrema

Last lesson