Everything about f, out of one picture AB & BC

Take $f(x) = 3x^{5} - 20x^{3}$, so $f'(x) = 15x^{2}(x^{2} - 4)$ and $f''(x) = 60x(x^{2} - 2)$.

Direction. $15x^{2} \ge 0$, so the sign of $f'$ is the sign of $x^{2} - 4$: positive outside $[-2, 2]$ and negative inside. So $f$ rises, falls, then rises, with a relative maximum at $x = -2$ where $f = 64$ and a relative minimum at $x = 2$ where $f = -64$. The critical point at $x = 0$ produces no sign change and therefore no extremum.

Concavity. $60x(x^{2} - 2)$ has three roots, at $-\sqrt{2}$, $0$, and $\sqrt{2}$, and the sign alternates across all three: down, up, down, up. Three inflection points.

So $x = 0$ is a critical point that is not an extremum and an inflection point, which is the combination worth being able to name. The two charts are built from different derivatives and split the line at different places, and a fact from one is never evidence for the other.

Reading a combined table means taking the columns one at a time. The $f'$ column answers rising and falling and locates the extrema. The $f''$ column answers bending and locates the inflection points. A row where both are negative says the function is falling and concave down, which are two independent descriptions of the same stretch of graph.

A single graph of $f'$ answers every structural question about $f$. Suppose the picture is $f'(x) = (x + 1)(x - 2)^{2}$.

1. Direction. Where the curve is below the axis $f$ falls, above it $f$ rises. Here it is negative on $(-\infty, -1)$ and positive after, so $f$ falls then rises.
2. Extrema. Crossings. At $x = -1$ the curve crosses from below, so $f$ has a relative minimum. At $x = 2$ it touches the axis and comes back up, so there is a critical point and no extremum.
3. Concavity. The rise and fall of this curve. It climbs to a peak at $x = 0$, falls to a low point at $x = 2$, then climbs again, so $f$ is concave up, then down, then up.
4. Inflection points. The peak and the low point of this curve, at $x = 0$ and $x = 2$.

Note what is not answerable: any value of $f$. Adding a constant to $f$ leaves this picture identical, so $f(0)$, the $y$-intercept, and the location of the $x$-intercepts are all invisible here. A question asking for $f(3)$ off a graph of $f'$ is asking for something the graph does not contain, unless one value of $f$ is supplied separately.

On the free-response section the conclusion is worth little and the reason is worth the point. A justification names the derivative, names what it does, and names where.

1. "$f$ has a relative maximum at $x = 3$ because $f'$ changes from positive to negative at $x = 3$."
2. "$f$ has an inflection point at $x = 1$ because $f''$ changes sign at $x = 1$."
3. "$f(5)$ is the absolute maximum on $[0, 5]$ because it is the largest of the candidate values $f(0)$, $f(2)$, and $f(5)$."

Three phrasings that do not earn the point, and why:

1. "because $f'(3) = 0$." True and insufficient: $x^{3}$ satisfies it at the origin with no extremum. Vanishing makes a candidate.
2. "because $f''(1) = 0$." Same shape of error one derivative up, and $x^{4}$ refutes it.
3. "because the graph turns around there." That restates the conclusion. A justification cites the derivative, not the shape it is being used to establish.

One more habit: name the function. "It changes from positive to negative" is ambiguous when three functions are in play, and the reader will not supply the missing word charitably.

Topic 5.7 established that $f'' = 0$ is not enough. In a synthesis problem that fact arrives disguised, because the data usually comes as a list of roots and the temptation is to count them.

1. A root without a change. If $f''(x) = 20x^{2}(x - 3)$, the roots are $0$ and $3$ and only $3$ is an inflection point, because the squared factor cannot change sign.
2. A change without a root. If $f''$ fails to exist at $c$ and flips sign across it, and $c$ is in the domain of $f$, that is an inflection point. $x^{1/3}$ at the origin is the standard case.
3. A change outside the domain. $\frac{1}{x}$ changes concavity across $x = 0$ and has no inflection point, because there is no point of the graph there.

The same three cautions in the language of a graph of $f'$: an inflection point of $f$ is a place where the graph of $f'$ turns around. A flat spot on the graph of $f'$ that keeps going the same way is not one, in exactly the way that a flat spot on the graph of $f$ is not an extremum.

Everything in this unit reduces to that one sentence, applied at the right level. A value tells you where to look. A sign tells you what is happening. A change of sign is what licenses a conclusion.