AP Calculus AB Concavity and the Second Derivative Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Concavity and the Second Derivative – Maths PDF

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This concavity test answers 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.

Use this answer key to score the concavity test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.

AP Calculus AB · Common Core State Standards: AP CALC AB 5.7, AP CALC AB 5.8, AP CALC AB 5.9

Concavity test answers: worked solutions

Question 1 – Concavity intervals [2 pts]

\(f^{\prime}=4x^{3}-24x\) and \(f^{\prime\prime}=12x^{2}-24=12(x^{2}-2)\).

Zeros: \(x=\pm\sqrt{2}\). Positive for \(|x| > \sqrt{2}\), negative between.

Final answer: concave up on (-∞, -√2) and (√2, ∞); concave down on (-√2, √2).

Scoring: 1 point for f″; 1 point for the intervals.

Question 2 – Inflection point [3 pts]

\(f^{\prime\prime}=6x-6\), zero at \(x=1\). \(f^{\prime\prime} < 0\) for \(x < 1\) and \(f^{\prime\prime} > 0\) for \(x > 1\). \(f(1)=1-3-9+1=-10\).

Final answer: \((1,-10)\).

Graph of f(x) = x³ − 3x² − 9x + 1.

Scoring: 1 point for f″ = 0; 1 point for the sign change; 1 point for the point.

Question 3 – Second derivative test [2 pts]

\(f^{\prime}=3x^{2}-3=0\) gives \(x=\pm 1\). \(f^{\prime\prime}=6x\).

\(f^{\prime\prime}(1)=6 > 0\): relative minimum, \(f(1)=0\). \(f^{\prime\prime}(-1)=-6 < 0\): relative maximum, \(f(-1)=4\).

Scoring: 1 point for the critical numbers; 1 point for the classification.

Question 4 – Graph of the derivative [3 pts]

\(f^{\prime}=\frac{x^{3}-4x}{2}\), so \(f^{\prime\prime}=\frac{3x^{2}-4}{2}\), zero at \(x=\pm\frac{2}{\sqrt{3}}\).

Relative minima at \(x=-2\) and \(x=2\) (\(f^{\prime}\) goes \(-\) to \(+\)); relative maximum at \(x=0\) (\(+\) to \(-\)).
Concave up where \(f^{\prime}\) increases: \(\left(-\infty,-\frac{2}{\sqrt{3}}\right)\) and \(\left(\frac{2}{\sqrt{3}},\infty\right)\).
\(x=\pm\frac{2}{\sqrt{3}}\approx\pm 1.155\).

The graph of f′ (not f).

Scoring: 1 point per part.

Question 5 – Reading a summary table [3 pts]

On \((-\infty,1)\) (\(f^{\prime} > 0\)).
At \(x=1\) (\(f^{\prime}\) changes from \(+\) to \(-\)).
At \(x=3\) (\(f^{\prime\prime}\) changes from \(-\) to \(+\)).

Scoring: 1 point per part.

Question 6 – Justify a statement [2 pts]

\(g^{\prime\prime}=12x^{2}\ge 0\) for all \(x\), so the sign of \(g^{\prime\prime}\) does not change at \(0\).

Final answer: No; there is no change of concavity, so there is no inflection point.

Scoring: 1 point for examining the sign of g″; 1 point for the conclusion.

Question 7 – Fastest growth [3 pts]

\(V^{\prime}=-3t^{2}+24t\) and \(V^{\prime\prime}=-6t+24\).
\(V^{\prime\prime}=0\) at \(t=4\), changing from positive to negative: the rate is maximal at t = 4 days.
\(V^{\prime}(4)=-48+96=48\) thousand gallons per day.

The rate V′(t).

Scoring: 1 point per part.

Question 8 – Finding constants [2 pts]

\(f^{\prime\prime}=6x+2a=0\) at \(x=1\) gives \(a=-3\). \(f^{\prime}(3)=27+6a+b=0\) gives \(27-18+b=0\), so \(b=-9\).

Final answer: \(a=-3,\ b=-9\).

Scoring: 1 point for a; 1 point for b.

Total score

Total

20 points

Key facts behind this concavity test answers

Key facts

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 test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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