AP Calculus AB Concavity and the Second Derivative Test with Point Values (PDF)

Find the intervals where \(f(x)=x^{4}-12x^{2}+5\) is concave up and concave down.

Find the inflection point of \(f(x)=x^{3}-3x^{2}-9x+1\) and justify with the sign of \(f^{\prime\prime}\).

Use the Second Derivative Test to classify the critical numbers of \(f(x)=x^{3}-3x+2\).

The graph shows \(f^{\prime}(x)=\frac{x(x-2)(x+2)}{2}\). (a) Where does \(f\) have relative extrema? (1 point) (b) On which intervals is \(f\) concave up? (1 point) (c) Find the \(x\)-coordinates of the inflection points. (1 point)

Use the table. (a) Where is \(f\) increasing? (1 point) (b) Where does \(f\) have a relative maximum? (1 point) (c) Where is there an inflection point? (1 point)

Jamal says that \(g(x)=x^{4}\) has an inflection point at \(x=0\) because \(g^{\prime\prime}(0)=0\). Is he correct? Explain.

The volume of water in a reservoir in thousands of gallons is \(V(t)=-t^{3}+12t^{2}\) for \(0\le t\le 9\) days. (a) Find \(V^{\prime}(t)\) and \(V^{\prime\prime}(t)\). (1 point) (b) When does the volume increase fastest? (1 point) (c) What is that maximum rate? (1 point)

The function \(f(x)=x^{3}+ax^{2}+bx\) has an inflection point at \(x=1\) and a relative minimum at \(x=3\). Find \(a\) and \(b\).