Rule

• Let $f$ be twice differentiable near $c$ and $f^{\mathrm{\prime }}(c)=0$.
• If $f^{\mathrm{\prime }\mathrm{\prime }}(c)<0$, then $f(c)$ is a local maximum (the curve is a cap at $c$).
• If $f^{\mathrm{\prime }\mathrm{\prime }}(c)>0$, then $f(c)$ is a local minimum (the curve is a cup at $c$).
• If $f^{\mathrm{\prime }\mathrm{\prime }}(c)=0$ the test fails; use the first-derivative test.

✍️ Worked examples

• Example 7 Use the second-derivative test on $f(x)=x^4-8x^2+10$.
• Solution:
• $f^{\mathrm{\prime }}(x)=4x^3-16x=4x(x-2)(x+2)$, so the critical points are $x=0$ and $x=\pm 2$.
• $f^{\mathrm{\prime }\mathrm{\prime }}(x)=12x^2-16$.
• f′′(0)=−16<0 local maximum f(0)=10.
• f′′(±2)=32>0 local minimum f(±2)=16−32+10=−6 at both points.
• Example 8 Find the extrema of $f(x)=x^3-3x^2-9x$.
• Solution:
• $f^{\mathrm{\prime }}(x)=3x^2-6x-9=3(x-3)(x+1)$; critical points $x=3$ and $x=-1$.
• $f^{\mathrm{\prime }\mathrm{\prime }}(x)=6x-6$.
• f′′(3)=12>0 local minimum f(3)=27−27−27=−27.
• f′′(−1)=−12<0 local maximum f(−1)=−1−3+9=5.

✍️ Worked examples

• Example 17:
• A silo is a vertical cylinder topped by a hemisphere.
• Its cylindrical part holds $108\pi$ m${}^3$ of grain.
• A unit area of the top costs twice a unit area of the wall.
• Find the radius and height that minimise the cost.
• Solution:
• Let the wall cost $c$ per m${}^2$ and the roof cost $2c$.
• Volume of the cylinder πr2h=108π, so h=108r2.
• Wall area $2\pi rh$; hemisphere area $2\pi r^2$.
• Cost $K=c(2\pi rh)+2c(2\pi r^2)=2\pi cr(h+2r)=2\pi c({\displaystyle \frac{108}{r}}+2r^2)$.
• $K^{\mathrm{\prime }}=2\pi c(-{\displaystyle \frac{108}{r^2}}+4r)=0$ gives $r^3=27$, so $r=3$.
• $K^{\mathrm{\prime }\mathrm{\prime }}=2\pi c({\displaystyle \frac{216}{r^3}}+4)>0$, so this is a minimum.
• Then h=1089=12 radius 3 m, cylinder height 12 m.
• Example 18:
• A 2 m high fence stands 2 m from a wall.
• Find the shortest ladder that rests on the ground, touches the top of the fence and leans on the wall.
• Solution:
• Let the foot of the ladder be $x$ m from the fence and the top be $y$ m up the wall. Similar triangles: ${\displaystyle \frac{y}{x+2}}={\displaystyle \frac{2}{x}}$, so $y={\displaystyle \frac{2(x+2)}{x}}$.
• Length squared L2=(x+2)2+y2=(x+2)2(1+4x2).
• Differentiate L2 d(L2)dx=2(x+2)(1+4x2)+(x+2)2(−8x3)=2(x+2)(x3−8)x3.
• It is zero at $x=2$ (since $x+2>0$) and changes from negative to positive there, a minimum.
• $L^2=16(1+1)=32$, so $L=4\sqrt{2}\approx 5.66$ m.

• Example 19 A line through $C(3,2)$ meets the axes at $A$ and $B$. Show that the least area of triangle $AOB$ is 12 square units.
• Solution:
• Let $A=(a,0)$ and $B=(0,b)$. The line passes through $C$: ${\displaystyle \frac{3}{a}}+{\displaystyle \frac{2}{b}}=1$, so $b={\displaystyle \frac{2a}{a-3}}$ with $a>3$.
• Area $S={\displaystyle \frac{1}{2}}ab={\displaystyle \frac{a^2}{a-3}}$.
• $S^{\mathrm{\prime }}={\displaystyle \frac{2a(a-3)-a^2}{(a-3{)}^2}}={\displaystyle \frac{a(a-6)}{(a-3{)}^2}}$.
• $S^{\mathrm{\prime }}=0$ at $a=6$, changing from negative to positive, so $S$ is least there.
• Then $b=4$ and $S={\displaystyle \frac{36}{3}}=12$ square units.
• Example 20 The sum of two positive numbers is 20. Find them if the sum of their squares is least.
• Solution:
• Let the numbers be $x$ and $20-x$; $S=x^2+(20-x{)}^2$.
• $S^{\mathrm{\prime }}=2x-2(20-x)=4x-40=0$ gives $x=10$; $S^{\mathrm{\prime }\mathrm{\prime }}=4>0$.
• Both numbers are 10 and the least sum of squares is 200.

• Why f”(c) = 0 does not automatically guarantee an inflection point: For f(x) = x⁴, f”(x) = 12x² so f”(0) = 0. However, 12x² ≥ 0 for all x ≠ 0 (no sign change, remaining concave up everywhere), so (0, 0) is a local minimum, not an inflection point.
• Optimum dimensions of a cylinder of fixed volume V to minimize total surface area: Area A = 2π r² + 2π r h. Since V = π r² h → h = V / (π r²), substituting gives A(r) = 2π r² + 2V/r. Setting dA/dr = 4π r – 2V/r² = 0 yields r³ = V / (2π), leading to h = 2r (height equals diameter).
• Systematic 7-step guide to curve sketching 1. Domain; 2. Intercepts (x and y); 3. Symmetry (even/odd/periodic); 4. Asymptotes; 5. Intervals of increase/decrease (f’); 6. Local extrema; 7. Concavity and inflection points (f”).