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Connecting a Function, Its First Derivative, and Its Second Derivative AB & BC

A combined table or a single derivative graph carries every structural fact about $f$, sorted by which column it comes from. For $f(x) = 3x^{5} - 20x^{3}$, the factor $15x^{2}$ in $f'$ keeps the sign tied to $x^{2} - 4$, giving a maximum at $x = -2$, a minimum at $x = 2$, and a critical point at the origin with no sign change; meanwhile $f'' = 60x(x^{2} - 2)$ alternates across all three of its roots, so there are three inflection points and the origin is one of them.

From a graph of $f'$ alone: position relative to the axis gives direction, crossings give extrema, the curve's own rise and fall gives concavity, and its turning points give inflection points. What such a graph never gives is a value of $f$, since adding a constant to $f$ leaves it unchanged. A justification has to name the derivative, what it does, and where: $f'$ changing from positive to negative, not $f'$ merely equalling zero, and $f''$ changing sign, not $f''$ merely vanishing.

f(x) = 3x⁵ − 20x³, f′ = 15x²(x² − 4), f″ = 60x(x² − 2) INTERVAL f′ f″ BEHAVIOUR OF f x < −2 POS NEG RISING, CONCAVE DOWN −2 < x < −1.414 NEG NEG FALLING, CONCAVE DOWN −1.414 < x < 0 NEG POS FALLING, CONCAVE UP 0 < x < 1.414 NEG NEG FALLING, CONCAVE DOWN 1.414 < x < 2 NEG POS FALLING, CONCAVE UP x > 2 POS POS RISING, CONCAVE UP AT x = −2: RELATIVE MAXIMUM 64. AT x = 2: RELATIVE MINIMUM −64. AT x = 0: f′(0) = 0 WITH NO SIGN CHANGE, SO NO EXTREMUM, AND f″ DOES CHANGE SIGN, SO IT IS AN INFLECTION POINT. THREE INFLECTIONS IN ALL.
Two columns, read separately. The rows where both are negative describe a stretch that is falling and concave down, which are independent facts about the same piece of curve.
THE CURVE IS f′(x) = (x + 1)(x − 2)². NOTHING BELOW IS A VALUE OF f. PEAK OF f′ f′ = 0, SIGN CHANGES f′ = 0, NO SIGN CHANGE f′ SHOWN. READING OFF f: MIN AT x = −1 (f′ CROSSES − TO +) NO EXTREMUM AT x = 2 (f′ TOUCHES) INFLECTION AT x = 0 (f′ PEAKS) INFLECTION AT x = 2 (f′ BOTTOMS) EVERY ANSWER ABOUT f IS IN THIS ONE PICTURE, AND NONE OF THEM IS A VALUE OF f.
Position gives direction, crossings give extrema, the curve's own rise and fall gives concavity, and its turning points give inflection points.

The work

3 ways in · any order
Lesson
Connecting a Function, Its First Derivative, and Its Second Derivative ›

Pulls the whole unit into one reading: a combined sign table taken column by column, a single graph of the derivative that answers every structural question about the function and no question about its values, the justification language the exam scores, and the three ways a claimed inflection point goes wrong.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items spanning the two failure modes of this topic: attributing a feature to the wrong one of a function, its derivative, and its second derivative, and counting roots of the second derivative as inflection points without confirming that the concavity changed.

Not yet available · 10 items
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Targeted Practice
Drill a single misconception ›

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.

Take the diagnostic to identify your misconceptions
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