Say which curve it is first AB & BC

Every feature of $f$ has three descriptions. Learning them as a table is worth more than learning them one at a time, because the errors are all substitutions of one row for another.

1. $f$ increasing $\;\leftrightarrow\;$ graph of $f'$ above the axis. Nothing follows about $f''$.
2. $f$ has a relative maximum $\;\leftrightarrow\;$ graph of $f'$ crosses the axis from $+$ to $-$ $\;\leftrightarrow\;$ $f'' < 0$ there, if $f''$ exists.
3. $f$ concave up $\;\leftrightarrow\;$ graph of $f'$ is rising $\;\leftrightarrow\;$ graph of $f''$ above the axis.
4. $f$ has an inflection point $\;\leftrightarrow\;$ graph of $f'$ has a relative extremum $\;\leftrightarrow\;$ graph of $f''$ crosses the axis.

Read the columns rather than the rows and the pattern shows up: a feature moves one column to the right by turning from a shape into a position, and from a position into a crossing. A turning point of $f$ becomes a crossing of $f'$. A turning point of $f'$ becomes a crossing of $f''$.

That is also where the confusion comes from. On a graph of $f'$, both "above the axis" and "rising" mean something, and they mean different things about $f$: one is direction and the other is concavity. Two readings of one curve, and choosing the wrong one produces a fluent, confident, wrong sentence.

Take $f(x) = x^{4} - 4x^{3}$, so $f'(x) = 4x^{2}(x - 3)$ and $f''(x) = 12x(x - 2)$.

1. Direction. $4x^{2} \ge 0$, so the sign of $f'$ is the sign of $x - 3$. Decreasing on $(-\infty, 3)$, increasing on $(3, \infty)$. One extremum: a relative minimum at $x = 3$, where $f(3) = -27$.
2. Concavity. $12x(x - 2)$ is positive, negative, positive across $0$ and $2$. Concave up, then down, then up, with inflection points at $(0, 0)$ and $(2, -16)$.
3. Anchors. $f(x) = x^{3}(x - 4)$, so the graph crosses the axis at $x = 0$ and $x = 4$.

The interesting point is $x = 0$. It is a critical point, since $f'(0) = 0$, and it is not an extremum, since $4x^{2}$ prevents the sign from changing. It is an inflection point, since $f''$ does change sign there. So the graph has a horizontal tangent at the origin while falling straight through it, and switches from concave up to concave down at the same moment.

Order the work this way every time: direction chart, concavity chart, then anchor points ($x$-intercepts, $y$-intercept, end behaviour). The two charts give the shape and the anchors give the position.

Given several curves on a shared $x$-axis and asked which is $f$, which is $f'$, and which is $f''$, one relationship does most of the work:

The zeros of the derivative sit under the turning points of the function.

So look for a curve whose $x$-intercepts line up vertically with another curve's peaks and valleys. The one with the intercepts is the derivative of the one with the peaks. Do it once to order two curves, then again to order the third.

Three more checks that catch the remaining ambiguity:

1. Count the turns. Differentiating usually removes one. A curve with two turning points is unlikely to be the derivative of a curve with one.
2. Check the direction. Where the candidate derivative is above the axis, the candidate function must be going up. One counterexample anywhere kills the pairing.
3. Use degree if they are polynomials. A parabola cannot be the derivative of a parabola.

What is not a valid check: which curve is drawn higher on the page, which is steeper, or which looks more complicated. None of those survive shifting a graph up by a constant, and shifting $f$ up by a constant does not change $f'$ at all.

Each of these is a row substitution from the table in section 1, and each appears constantly.

1. "The graph of $f'$ has a maximum at $x = 2$, so $f$ has a maximum at $x = 2$." A maximum of $f'$ is where $f$ climbs fastest. What it actually marks on $f$ is an inflection point, since $f'$ turning around is $f''$ crossing zero.
2. "The graph of $f'$ is above the axis, so $f$ is concave up." Above the axis is direction. Concavity is whether that curve is rising.
3. "The graph shown crosses the axis at $x = 3$, so $f(3) = 0$." Only if the graph shown is $f$. If it is $f'$, then $f'(3) = 0$ and $f(3)$ could be anything at all.
4. "$f''$ is increasing, so $f$ is concave up." Two derivatives off. $f''$ increasing is a claim about $f'''$; concavity is the sign of $f''$.

One habit prevents all four. Before reading anything off a curve, say out loud which function it is, and then say which feature of that function answers the question. The picture is almost never wrong. The sentence attached to it is.