Determining Concavity of Functions over Their Domains AB & BC
A function is concave up exactly where $f'' > 0$, which is the same as saying $f'$ is increasing, and concave down where $f'' < 0$. Geometrically, a concave up graph lies above every one of its tangent lines. Direction and concavity are independent: $e^{x}$ is increasing and concave up, $\sqrt{x}$ is increasing and concave down, $e^{-x}$ is decreasing and concave up, and $-x^{2}$ on $x > 0$ is decreasing and concave down. Direction asks whether the slope is positive; concavity asks whether the slope is growing.
The concavity chart is built exactly like the direction chart, one derivative higher, and it is split by the zeros of $f''$, the points where $f''$ fails, and the gaps in the domain of $f$. For $x^{3} - 3x^{2} - 9x + 5$ the direction chart splits at $-1$ and $3$ while the concavity chart splits at $1$, which is why one cannot be read off the other. On a graph of $f'$, above and below the axis answers the direction question and rising and falling answers the concavity question.
The work
3 ways in · any order
Lesson
Determining Concavity of Functions over Their Domains
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Ties concave up to a positive second derivative, to a rising first derivative, and to a graph sitting above its own tangent lines, then puts all four pairings of direction with concavity on one page so neither can be inferred from the other, builds the concavity chart with its own marks, and reads bending off the shape of a derivative graph.
Diagnostic
10-item topic check
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Ten items spanning the two failure modes of this topic: deciding concavity from the sign of the first derivative or from the assumption that increasing means curving upward, and building the chart from the wrong function or missing the points that split it.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.