Maxima and minima of single-variable functions

At a Glance

• Frequency: 4 sub-parts across 4 of 13 years (2013, 2015, 2021, 2022)
• Priority tier: T3
• Marks (count): 10 (2), 15 (2)
• Average solve time: ~8 min
• Difficulty mix: easy 2, medium 2
• Section: A | Dominant type: computation

Why This Chapter Matters

These questions appear in Section A and test two flavours: (a) Hessian-based extrema of two-variable functions (find critical points, classify via $D=f_{xx}f_{yy}-f_{xy}^2$), and (b) extrema of one-variable functions on a closed interval. The Hessian test sometimes produces a degenerate case ($D=0$) that requires direct line-slice analysis — this is the key difficulty. The closed-interval case is purely algorithmic.

Minimum Theory

Critical points. Set $f'(x)=0$ (for 1D) or $\nabla f=\mathbf{0}$ (for 2D) and solve.

Second derivative test (1D). At a critical point $x_0$: if $f''(x_0)>0$, local min; if $f''(x_0)<0$, local max; if $f''(x_0)=0$, inconclusive (use higher derivatives or direct comparison).

Closed interval (1D). On $[a,b]$: evaluate $f$ at all critical points inside $(a,b)$ and at the endpoints $a$, $b$. The largest value is the absolute max, the smallest is the absolute min.

Second derivative test (2D). At a critical point $(x_0,y_0)$ of $f(x,y)$: $D = f_{xx}f_{yy} - f_{xy}^2.$

• $D>0$, $f_{xx}>0$: local minimum.
• $D>0$, $f_{xx}<0$: local maximum.
• $D<0$: saddle point.
• $D=0$: inconclusive — examine $f$ along line slices through the critical point.

Degenerate case $D=0$. Evaluate $f$ along two lines through the critical point. If $f$ exceeds the critical value along one line and falls below it along another, the point is a saddle.

Question Archetypes

hessian-extrema (3 question(s); 2013, 2015, 2022)

Recognition Cues — “Find max/min values of $f(x,y)$”; no constraint given (unconstrained); function is polynomial or similar.

Solution Template

1. Compute $f_x$, $f_y$; solve $f_x=0$ and $f_y=0$ simultaneously.
2. For each critical point compute $f_{xx}$, $f_{yy}$, $f_{xy}$, and $D=f_{xx}f_{yy}-f_{xy}^2$.
3. Classify each point; for $D=0$ use line slices.
4. State the extreme values (and note whether a global max/min exists).

Worked Example

2013 Paper 2, 2013-P2-Q3c (10 marks)

Find max/min of $f(x,y)=y^2+4xy+3x^2+x^3+1$.

Step 1. $f_x=6x+4y+3x^2=0$, $f_y=2y+4x=0\Rightarrow y=-2x$. Substitute: $6x-8x+3x^2=0\Rightarrow x(3x-2)=0$. Critical points: $(0,0)$ and $(2/3,-4/3)$.

Step 2. $f_{xx}=6+6x$, $f_{yy}=2$, $f_{xy}=4$.

At $(0,0)$: $D=(−4)2−16=−4<0$ → saddle.

At $(2/3,-4/3)$: $f_{xx}=10$, $D=10\cdot2-16=4>0$, $f_{xx}>0$ → local min, $f=23/27$.

Step 3. No global max (as $x\to\infty$, $f\to\infty$ via the $x^3$ term).

$\boxed{\text{Local min at }(2/3,-4/3)\text{ with value }23/27;\;\text{saddle at }(0,0).}$

Degenerate case example (2016): For $f=x^4+y^4-2x^2+4xy-2y^2$, the critical point $(0,0)$ yields $D=0$. Slices: $f(t,t)=2t^4\ge0$ (rises), $f(t,-t)=2t^4-8t^2<0$ for small $t$ (falls) — saddle.

Common Traps

• $D=0$ is inconclusive; don’t guess. Showing one line where $f$ increases and one where it decreases is the required argument.
• Global vs local. The presence of odd-degree terms (like $x^3$) means $f$ is unbounded; no global max or min may exist even if local extrema do.

extrema-on-interval (1 question(s); 2021)

Worked Example

2021 Paper 2, 2021-P2-Q2a (15 marks)

Find max and min of $f(x)=x^3-9x^2+26x-24$ on $[0,1]$.

$f'(x)=3x^2-18x+26$. Discriminant: $324-312=12>0$, roots at $3\pm1/\sqrt{3}\approx2.42$ and $3.58$ — both outside $[0,1]$. So $f'(x)>0$ on $[0,1]$ (check: $f'(0)=26>0$) — $f$ is strictly increasing.

Absolute max at $x=1$: $f(1)=1-9+26-24=-6$. Absolute min at $x=0$: $f(0)=-24$.

$\boxed{f_{\max}=-6\text{ at }x=1;\quad f_{\min}=-24\text{ at }x=0.}$

Common Traps

• Check critical points are inside the interval. When no critical points lie inside $(a,b)$, the function is monotone on $[a,b]$ and the extrema are at the endpoints.
• Don’t confuse the discriminant sign. $\Delta=324-312=12>0$ means two real roots; the roots are near $2.4$ and $3.6$, both outside $[0,1]$.

Marks-Aware Writing

For the Hessian test: write $f_x$, $f_y$ explicitly; show the system solution step; build a table of $(f_{xx}, f_{yy}, f_{xy}, D)$ at each critical point; classify each in words. For $D=0$, exhibit two specific line slices. Marking: roughly 2 marks per critical point’s classification, plus 2 marks for the final statement.

For extrema on interval: critical points from $f'=0$; check each is in the interval (state this); evaluate $f$ at the critical points and endpoints; state the max/min with their locations. On a 15-mark question, more detail is expected — show the discriminant calculation, check sign of $f'$ on the interval.

Practice Set

• 2022 question (see atom_stats) — additional Hessian computation on a quartic.
• 2015 Paper 2 Hessian question — check analysis/solutions/ for details.