Curve Sketching Practice

These questions follow Motion on a Line and Curve Sketching — full sketches from equations, reading functions from their derivatives, and motion analysed with the same sign charts. Run the checklist; let the chart do the remembering.

From equation to sketch

1. Analyse and sketch $f(x)=x^3-6x^2+9x$: intercepts, intervals of increase and decrease, local extremes, concavity, and any point of inflection. Verify with technology afterward.
2. For $f(x)=x^4-4x^3$, determine all critical numbers and classify each, then find any points of inflection.

Answer 1

Intercepts: $f(x)=x(x-3{)}^2$, roots at $x=0$ and $x=3$ (double). First derivative: $f^{\prime }(x)=3x^2-12x+9=3(x-1)(x-3)$, zero at 1 and 3. Sign chart: $+$, $-$, $+$ — increasing, decreasing, increasing. Local maximum $(1,4)$; local minimum $(3,0)$. Second derivative: $f^{\prime \prime }(x)=6x-12$, zero at $x=2$: concave down before, up after — inflection point $(2,2)$. Note the corroboration: the double root at 3 and the minimum at $(3,0)$ are the same fact — the graph touches the axis without crossing.

Answer 2

$f^{\prime }(x)=4x^3-12x^2=4x^2(x-3)$: critical numbers 0 and 3. Sign of $f^{\prime }$: negative for $x<0$, still negative for $0<x<3$, positive after 3. No sign change at 0 — not an extreme, just a flat pause mid-descent; sign change $-$ to $+$ at 3 — local (indeed global) minimum $(3,-27)$. $f^{\prime \prime }(x)=12x^2-24x=12x(x-2)$: zero at 0 and 2 with genuine sign changes, so inflection points $(0,0)$ and $(2,-16)$. The critical number that was not an extreme turns out to be an inflection point with a horizontal tangent.

Reading derivatives

3. Determine $f^{\prime \prime }(x)$ for the simple rational function $f(x)=\frac{1}{x}$, and state what it says about concavity.
4. The derivative of $f(x)$ is $g(x)=(x-1)(x-3)$, and $f(0)=0$. Describe the key features of $f$, determine its equation, and explain what changes if instead $f(0)=2$.
5. You are told only that $f^{\prime }(x)>0$ for $x<2$, $f^{\prime }(2)=0$, and $f^{\prime }(x)<0$ for $x>2$. What must the graph of $f$ look like — and why are infinitely many different graphs consistent with this information?

Answer 3

$f(x)=x^{-1}$, so $f^{\prime }(x)=-x^{-2}$ and $f^{\prime \prime }(x)=2x^{-3}=\frac{2}{x^3}$. For $x>0$, $f^{\prime \prime }>0$: concave up (the right branch holds water). For $x<0$, $f^{\prime \prime }<0$: concave down. The two branches bend opposite ways, which the graph confirms at a glance.

Answer 4

$f^{\prime }=g$ is an upward parabola, positive–negative–positive around its roots: $f$ increases to a local maximum at $x=1$, decreases to a local minimum at $x=3$, then increases. Undoing $g(x)=x^2-4x+3$ term by term (each power steps back up): $f(x)=\frac{x^3}{3}-2x^2+3x+C$, and $f(0)=0$ forces $C=0$. Features: maximum $\left(1,\frac{4}{3}\right)$, minimum $(3,0)$, inflection at $x=2$ where $f^{\prime \prime }(x)=2x-4$ changes sign. If $f(0)=2$, only $C$ changes: the whole graph rides up two units — same shape, same $x$-locations for every feature.

Answer 5

$f$ climbs, flattens exactly once at $x=2$, then falls: one local maximum at $x=2$ and no other features are forced. But the information is all about slopes, and slopes are blind to height — any vertical translation of a consistent graph is another consistent graph. Infinitely many answers, one shape. Knowing $f^{\prime }$ everywhere pins down everything about $f$ except where it sits.

Motion as sketching

6. A particle moves along a line with position $s(t)=t^3-9t^2+24t$ metres after $t$ seconds, $t\ge 0$. When is the particle at rest? When is it moving in the positive direction? On which intervals is it speeding up?

Answer 6

Velocity: $v(t)=3t^2-18t+24=3(t-2)(t-4)$ — at rest at $t=2$ and $t=4$. Sign chart: positive on $0\le t<2$, negative on $2<t<4$, positive for $t>4$ — moving forward, backing up, moving forward again. Acceleration: $a(t)=6t-18$, negative before $t=3$, positive after. Speeding up where $v$ and $a$ agree in sign: on $2<t<3$ (both negative) and $t>4$ (both positive). Slowing down on $0\le t<2$ and $3<t<4$. The sign chart from Curve Sketching and the agreement table from Motion on a Line are the same tool wearing different labels.