Chapter 9 — Key Takeaways

A compact recap of everything in "Applications of Derivatives." The one sentence to carry away: $f'$ governs slope (increase, decrease, extrema); $f''$ governs curvature (concavity, inflection); the Mean Value Theorem makes all of it rigorous.

Extrema: Absolute vs. Local (§9.2–9.3)

• Absolute (global) maximum at $c$: $f(c) \ge f(x)$ for every $x$ in the domain. Local maximum: $f(c) \ge f(x)$ only for $x$ near $c$. (Minima reverse the inequality.)
• Extreme Value Theorem: a function continuous on a closed, bounded interval $[a,b]$ attains both an absolute max and an absolute min. Both hypotheses (continuity, closed interval) are load-bearing.
• Critical point: a domain value $c$ where $f'(c) = 0$ or $f'(c)$ does not exist. Both cases matter — corners and cusps (like $|x|$ at $0$) are critical points too.
• Fermat's Theorem: an interior local extremum with a derivative forces $f'(c) = 0$. One-way only: $f'(c) = 0$ is necessary, not sufficient (counterexample $x^3$ at $0$).
• Closed Interval Method for absolute extrema on $[a,b]$: (1) find interior critical points, (2) evaluate $f$ there and at both endpoints, (3) largest value is the max, smallest is the min.

Monotonicity (§9.4)

• Increasing/Decreasing Test: on an interval, $f' > 0 \Rightarrow$ increasing; $f' < 0 \Rightarrow$ decreasing; $f' = 0 \Rightarrow$ constant. (Proved from the MVT in §9.7.3.)
• The sign chart of $f'$ — split the line at critical points, test one point per interval — is the single most useful diagram in differential calculus.

The First Derivative Test (§9.5)

At a critical point $c$, examine the sign of $f'$ on each side:

Always works — even at corners where $f''$ does not exist — and never returns "inconclusive."

Concavity and the Second Derivative Test (§9.6)

• Concave up ($f'' > 0$): graph lies above its tangents — a bowl/smile; slope $f'$ is increasing.
• Concave down ($f'' < 0$): graph lies below its tangents — a dome/frown; slope $f'$ is decreasing.
• Second Derivative Test at a critical point with $f'(c) = 0$:

• Inflection point: where concavity changes — $f''$ changes sign. $f''(c) = 0$ is necessary but not sufficient ($x^4$ at $0$: $f''=0$ but no sign change, hence no inflection). An inflection can also occur where $f''$ fails to exist, provided concavity genuinely flips.

The Mean Value Theorem (§9.7)

• MVT: if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c \in (a,b)$ with $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$ Somewhere inside, the instantaneous rate equals the average rate (tangent parallel to secant).
• Rolle's Theorem is the special case $f(a) = f(b)$: then some interior $c$ has $f'(c) = 0$. It is the engine of the MVT proof.
• Proof chain (the chapter's spine): Extreme Value Theorem $\Rightarrow$ Fermat $\Rightarrow$ Rolle $\Rightarrow$ MVT. The MVT proof "tilts" Rolle by subtracting the secant line $g(x) = f(x) - L(x)$.
• Consequences of the MVT (§9.7.3): the Increasing/Decreasing Test; the Constant Function Theorem ($f' \equiv 0 \Rightarrow f$ constant); and "equal derivatives differ by a constant" — the fact that justifies the $+C$ in every antiderivative and underlies the Fundamental Theorem of Calculus (Chapter 14).

L'Hôpital's Rule (§9.8)

• Applies only to $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$. Then $\lim \frac{f}{g} = \lim \frac{f'}{g'}$ provided the right-hand limit exists.
• Other indeterminate forms must be reshaped first:
• $0 \cdot \infty$: rewrite as a quotient $\frac{f}{1/g}$.
• $\infty - \infty$: combine over a common denominator.
• $0^0,\ 1^\infty,\ \infty^0$: take $\ln$, evaluate, then exponentiate.
• Key growth facts: $e^x$ outpaces every polynomial; every polynomial outpaces $\ln x$.
• Two traps (§9.8.3): never apply L'Hôpital to a determinate form (e.g. $\tfrac{2}{2}$); and the rule fails when the ratio of derivatives has no limit (e.g. $\frac{x+\sin x}{x}$) — the original limit may still exist.

The Curve-Sketching Procedure (§9.9–9.10)

1. Domain — exclude division-by-zero, even roots of negatives.
2. Intercepts — $y$-intercept $f(0)$; $x$-intercepts solve $f(x) = 0$ (test small integers for exact roots first).
3. Symmetry — even ($f(-x)=f(x)$), odd ($f(-x)=-f(x)$), or periodic.
4. Asymptotes — vertical (denominator zeros not cancelled), horizontal ($\lim_{x\to\pm\infty}f$), oblique (when $\deg p = \deg q + 1$, via long division).
5. First derivative — critical points, sign chart, intervals of increase/decrease.
6. Local extrema — classify with the first or second derivative test.
7. Second derivative — sign chart for concavity, locate inflection points.
8. Assemble — plot special points; connect honoring slope and concavity.

Common Errors to Avoid

• Solving only $f'(x) = 0$ and missing critical points where $f'$ is undefined (corners, cusps).
• Treating $f'(c) = 0$ as a guarantee of an extremum (it is only a candidate — recall $x^3$).
• Calling $f''(c) = 0$ an "inflection point" automatically — it requires a sign change of $f''$.
• Applying L'Hôpital to a non-indeterminate form, or trusting it when the derivative-ratio limit fails to exist.
• Forgetting that endpoints compete with critical points in the Closed Interval Method.
• Rounding to a numerical root when an exact root (e.g. $x = 1$) factors the whole problem.

Connections

• Backward: the derivative rules of Chapters 6–8 are the inputs; the limit machinery of Chapters 3–4 underlies the EVT and L'Hôpital.
• Forward: Chapter 10 turns critical-point analysis loose on optimization word problems; Chapter 11 uses tangent lines and concavity for linear approximation and Newton's method; the MVT returns in Chapter 14 as the keystone of the Fundamental Theorem of Calculus, and again (carried to higher order) as Taylor's theorem in Chapter 23.
• Across fields (§9.12): equilibria of potential energy (physics), marginal-revenue-equals-marginal-cost (economics), pharmacokinetic peaks (biology), and the convexity of a loss surface (data science) are all the same $f'$-and-$f''$ analysis in different clothing.

The threshold idea. A function's shape is written in its derivatives. Learn to read $f'$ and $f''$ together and you can reconstruct any curve — peaks, valleys, bends, and transitions — without plotting it by brute force.