Connecting a Function, Its First Derivative, and Its Second Derivative
Introduced| English | Meaning |
|---|---|
| zero/ˈzɪərəʊ/ | An input value at which a function's output equals zero. |
Three graphs telling one story
- $f$, $f'$, and $f''$ are three views of the same function — and they must all agree.
- A feature of $f$ shows up as a specific feature of $f'$ or $f''$.
- The exam loves handing you one of the three and asking about another.
- Master the translation table and these questions become quick.
What the zeros of $f'$ say
- A zero 零点 of $f'$ (where $f'=0$) is a critical point of $f$ — a possible extremum.
- $f'$ goes $+\to-$ at that zero → $f$ has a local max; $-\to+$ → local min.
- Where $f'>0$, $f$ increases; where $f'<0$, $f$ decreases.
- So the sign of $f'$ gives increase/decrease, and its zeros give extrema.
A cubic with its first and second derivatives on the same x-axis
One function, three readings
y = ax³ + cx
On this curve, the zeros of $f'$ mark the peak and valley, and the zero of $f''$ marks the inflection.
A sign-changing zero of $f'$ marks, on the graph of $f$, a...
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Zeros of $f'$ (sign-changing) are extrema of $f$.
On an interval where the graph of $f'$ is positive, $f$ is increasing.
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$f'>0$ ⇔ $f$ increasing.
What the zeros of $f''$ say
- A sign-changing zero of $f''$ is a point of inflection of $f$.
- Where $f''>0$, $f$ is concave up; where $f''<0$, concave down.
- Note: $f''$ is the derivative of $f'$, so a zero of $f''$ is where $f'$ has a max or min (its steepest slope on $f$).
- The sign of $f''$ gives concavity; its sign-changing zeros give inflections.
A sign-changing zero of $f''$ marks, on the graph of $f$, a...
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Sign-changing zeros of $f''$ are inflection points of $f$.
From the graph of $f''$ you can read which features of $f$?
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Concavity + inflections come from $f''$; exact values need $f$ itself.
Reading across representations
- Given a graph of $f'$: its zeros → extrema of $f$; its sign → increase/decrease of $f$; its own increase/decrease → concavity of $f$.
- Given a table of values: estimate $f'$ and $f''$ with difference quotients, then apply the same rules.
- Given a formula: differentiate and use sign charts.
- Same logic, three languages — translate freely.
The graph of $f'$ is positive then negative, crossing zero at $x=2$. At $x=2$, $f$ has a...
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$f'$ goes $+\to-$ → local max of $f$.
On a graph of $f'$, it is the ____ (not the peaks) that mark the extrema of $f$.
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Zeros of $f'$ = extrema of $f$; peaks of $f'$ relate to inflections of $f$.
Keep the levels straight. A zero of $f'$ is an extremum of $f$ (not of $f'$). A zero of $f''$ is an inflection of $f$ and an extremum of $f'$. Reading a graph of $f'$ as if it were $f$ — treating its peaks as maxima of $f$ — is a classic mix-up. Its zeros, not its peaks, mark $f$'s extrema.
You are given the graph of $f'$, which is positive on $(-\infty,2)$, zero at $x=2$, and negative after. What happens to $f$ at $x=2$?
- $f'$ changes $+\to-$ at $x=2$ → $f$ has a local maximum there.
- $f$ is increasing before $x=2$ and decreasing after (matching $f'>0$ then $f'<0$).
- The value $f'(2)=0$ means a horizontal tangent on $f$.
$f$, $f'$, $f''$ tell one consistent story. Zeros of $f'$ (sign-changing) are $f$'s extrema; the sign of $f'$ gives increase/decrease. Zeros of $f''$ (sign-changing) are $f$'s inflection points; the sign of $f''$ gives concavity. Translate this across graphs, tables, and formulas.