Chapter 9 — Quiz
10 questions covering critical points, the derivative tests, concavity and inflection, the Mean Value Theorem, L'Hôpital's rule, and curve sketching. Try each before opening the answer. Each answer cites the index section where the idea is developed.
1. Which of the following is a critical point of $f(x) = x^{2/3}$, and why?
Answer
$x = 0$. Here $f'(x) = \tfrac23 x^{-1/3}$ is **undefined** at $x = 0$ (a cusp), and a critical point is any domain value where $f' = 0$ *or* $f'$ does not exist. This is exactly the case students miss when they only solve $f'(x) = 0$. *(§9.3)*2. Find the absolute maximum and minimum of $f(x) = x^3 - 3x$ on $[0, 2]$.
Answer
$f'(x) = 3x^2 - 3 = 3(x-1)(x+1)$; the only interior critical point in $[0,2]$ is $x = 1$. Candidates: $f(0) = 0$, $f(1) = -2$, $f(2) = 2$. Absolute max $= 2$ at $x = 2$; absolute min $= -2$ at $x = 1$. (Endpoints compete on equal footing — the Closed Interval Method.) *(§9.3)*3. State the intervals on which $f(x) = x^4 - 2x^2$ is increasing.
Answer
$f'(x) = 4x^3 - 4x = 4x(x-1)(x+1)$. Sign chart: $f' < 0$ on $(-\infty,-1)$, $f'>0$ on $(-1,0)$, $f'<0$ on $(0,1)$, $f'>0$ on $(1,\infty)$. **Increasing on $(-1, 0)$ and $(1, \infty)$.** *(§9.4)*4. A critical point $c$ has $f'(c) = 0$ and $f''(c) = 0$. What, if anything, can you conclude about whether $c$ is a local extremum?
Answer
**Nothing** from the second derivative test — it is inconclusive when $f''(c) = 0$. You must fall back on the first derivative test (check the sign change of $f'$). Example: $x^4$ has $f'(0)=f''(0)=0$ yet a clear minimum; $x^3$ has $f'(0)=f''(0)=0$ yet no extremum. *(§9.6)*5. Find the inflection points of $f(x) = x^4 - 6x^2$.
Answer
$f''(x) = 12x^2 - 12 = 12(x-1)(x+1)$, which changes sign at $x = \pm 1$. Since $f(\pm1) = 1 - 6 = -5$, the inflection points are $(-1, -5)$ and $(1, -5)$. *(§9.6)*6. State the two hypotheses of the Mean Value Theorem and give its conclusion in one sentence.
Answer
Hypotheses: $f$ is **continuous on $[a,b]$** and **differentiable on $(a,b)$**. Conclusion: there exists $c \in (a,b)$ with $f'(c) = \frac{f(b)-f(a)}{b-a}$ — i.e. somewhere inside, the instantaneous rate of change equals the average rate of change (the tangent is parallel to the secant). *(§9.7)*7. Rolle's theorem fails to apply to $f(x) = |x|$ on $[-1, 1]$ even though $f(-1) = f(1)$. Which hypothesis fails, and what does this show?
Answer
**Differentiability** fails — $f'(0)$ does not exist (corner at the origin). Despite $f(-1)=f(1)=1$, there is no point where $f'=0$. This shows the differentiability hypothesis is genuinely load-bearing, not decorative. *(§9.7.1)*8. Evaluate $\displaystyle\lim_{x \to 0} \frac{x - \sin x}{x^3}$.
Answer
Form $\tfrac00$. Apply L'Hôpital three times, each stage still $\tfrac00$: $\frac{1-\cos x}{3x^2} \to \frac{\sin x}{6x} \to \frac{\cos x}{6} = \frac16$. The answer is $\boxed{\tfrac16}$. *(§9.8.1)*9. Why can't you apply L'Hôpital's rule directly to $\displaystyle\lim_{x \to 0^+} x \ln x$, and how do you fix it?
Answer
It is a $0 \cdot (-\infty)$ form, and L'Hôpital applies *only* to $\tfrac00$ or $\tfrac{\infty}{\infty}$. Rewrite as a quotient: $\frac{\ln x}{1/x}$ (form $\tfrac{-\infty}{\infty}$), then L'Hôpital gives $\frac{1/x}{-1/x^2} = -x \to 0$. So the limit is $0$. *(§9.8.2)*10. In the full curve-sketching procedure, what role does the second derivative play that the first derivative cannot?
Answer
The second derivative reports **concavity** — whether the curve bends upward (bowl, $f''>0$) or downward (dome, $f''<0$) — and locates **inflection points** where the concavity changes ($f''$ changes sign). The first derivative only gives slope (increase/decrease and the location of extrema); it cannot distinguish a concave-up rise from a concave-down rise. Together $f'$ and $f''$ pin down the shape completely. *(§9.6, §9.9)*Scoring Guide
| Score | Interpretation |
|---|---|
| 9–10 | Excellent. You can read a function's shape fluently from its derivatives and handle the MVT and L'Hôpital with care. Ready for optimization (Chapter 10). |
| 7–8 | Solid. Review the one or two ideas you missed — most commonly the inconclusive second-derivative case (Q4) or the indeterminate-form bookkeeping (Q9). |
| 5–6 | Partial. Re-read §9.5–9.6 (the derivative tests) and §9.8 (L'Hôpital), then redo the exercises in Parts C and E. |
| 0–4 | Revisit the chapter from §9.3. Focus on the sign-chart method (§9.4) — it is the single most useful diagram in the chapter, and most other tools build on it. |
Most-missed question: Q4. The reflex to call $f''(c)=0$ an "inflection point" or "no extremum" is the chapter's central trap (§9.6). When the second derivative test is silent, the first derivative test still speaks — use it.