Chapter 31 — Exercises

38 problems on critical points, Hessian classification, absolute extrema on closed regions, Lagrange multipliers, and applications. Tiers ⭐ to ⭐⭐⭐⭐.

Work these by hand first; the Hessian discriminant $D = f_{xx}f_{yy} - f_{xy}^2$ and the Lagrange condition $\nabla f = \lambda\nabla g$ are the two workhorses, so make each automatic. A sympy or scipy check (Section 31.4, Section 31.11) is a good final step, but the understanding lives in the hand computation.

Difficulty tiers: ⭐ routine · ⭐⭐ standard · ⭐⭐⭐ challenging · ⭐⭐⭐⭐ deep/multi-step.

Part A — Finding Critical Points (Section 31.2)

A1. ⭐ Find all critical points of $f(x,y) = x^2 + y^2 - 4x + 6y + 1$.

A2. ⭐ Find all critical points of $f(x,y) = 3x^2 + 2y^2 - 12x + 8y$.

A3. ⭐ Find the single critical point of $f(x,y) = x^2 - xy + y^2 - 3y$.

A4. ⭐ Show that $f(x,y) = e^{-(x^2+y^2)}$ has exactly one critical point and state its location.

A5. ⭐⭐ Find all critical points of $f(x,y) = x^3 - 3x + y^2$. (There are two.)

A6. ⭐⭐ Find all real critical points of $f(x,y) = x^3 + y^3 - 3xy + 5$. Confirm there are exactly two, mirroring Example 2 of Section 31.4.

A7. ⭐⭐ Find all critical points of $f(x,y) = \sin x + \sin y + \sin(x+y)$ on the open square $0 < x,y < \pi$.

A8. ⭐ A student solves $f_x = 0$ to get $x = 2$ and separately solves $f_y = 0$ to get $y = -1$, then declares $(2,-1)$ critical. Referencing the Common Pitfall in Section 31.4, explain when this reasoning fails and when it happens to be valid.

Part B — Classifying with the Hessian (Sections 31.3–31.4)

B1. ⭐ For $f(x,y) = x^2 + 3y^2$, compute the Hessian, the discriminant $D$, and classify the critical point at the origin.

B2. ⭐ Classify the critical point of $f(x,y) = 4 - x^2 - y^2$ at the origin using the discriminant test.

B3. ⭐ Classify the origin for $f(x,y) = x^2 - y^2$ (the model saddle of Section 31.2).

B4. ⭐⭐ Classify every critical point of $f(x,y) = x^2 - 4xy + y^2$. (Watch the cross term: $f_{xy} \ne 0$.)

B5. ⭐⭐ Classify the critical points of $f(x,y) = x^3 - 3x + y^2$ from A5.

B6. ⭐⭐ Classify the critical points of $f(x,y) = x^3 + y^3 - 3xy + 5$ from A6.

B7. ⭐⭐ For $f(x,y) = x^2 y - x^2 - y^2$, find and classify all critical points.

B8. ⭐⭐ Find and classify all critical points of $f(x,y) = 2x^3 - 6x + 3y^2 - 6y$.

B9. ⭐⭐ Show that $f(x,y) = x^4 + y^4$ has a critical point at the origin where $D = 0$, then determine its type directly (as in Example 3 of Section 31.4). What kind of point is it?

B10. ⭐⭐⭐ For $f(x,y) = x^2 y^2$, confirm $D = 0$ at the origin and classify by inspecting the function along the axes and along $y = x$.

B11. ⭐⭐⭐ The function $f(x,y) = (y - x^2)(y - 2x^2) = y^2 - 3x^2 y + 2x^4$ has a critical point at the origin with $D = 0$. Show the origin is a minimum along every straight line through it, yet is not a local minimum of $f$ (test the parabolic path $y = \tfrac{3}{2}x^2$). This famous example shows why path-testing must include curves, not just lines.

B12. ⭐⭐⭐ Use the eigenvalue criterion of Section 31.6 to classify the critical point of $f(x,y,z) = x^2 + y^2 + z^2 - 2xy$ at the origin. Find the Hessian's eigenvalues; the determinant trick does not apply in three variables.

Part C — Absolute Extrema on Closed Regions (Section 31.5)

C1. ⭐⭐ Find the absolute maximum and minimum of $f(x,y) = x^2 + y^2 - 2x$ on the closed disk $x^2 + y^2 \le 4$.

C2. ⭐⭐ Find the absolute extrema of $f(x,y) = x + y$ on the closed square $0 \le x \le 1$, $0 \le y \le 1$.

C3. ⭐⭐⭐ Find the absolute extrema of $f(x,y) = xy - x - y + 3$ on the triangle with vertices $(0,0)$, $(2,0)$, $(0,2)$. Check the interior, all three edges, and all three corners.

C4. ⭐⭐⭐ Find the absolute extrema of $f(x,y) = x^2 - 2xy + 2y^2$ on the closed disk $x^2 + y^2 \le 8$.

C5. ⭐⭐⭐ Find the absolute maximum of $f(x,y) = 2x^2 + y^2$ on the closed region bounded by $y = x^2$ and $y = 4$. Sketch the region first.

C6. ⭐⭐⭐ Find the warmest and coldest points if the temperature on the closed unit disk $x^2 + y^2 \le 1$ is $T(x,y) = x^2 + 2y^2 - x$.

C7. ⭐⭐ A continuous $f$ on a closed, bounded region $R$ has no interior critical points. Where must its absolute extrema occur, and why? Cite the Extreme Value Theorem (Section 31.5) and contrast it with the one-dimensional version from Chapter 9.

Part D — Lagrange Multipliers, Single Constraint (Section 31.8)

D1. ⭐⭐ Use Lagrange multipliers to find the extrema of $f(x,y) = x^2 + y^2$ subject to $x + 2y = 5$. Interpret the result as the squared distance from the origin to a line.

D2. ⭐⭐ Maximize $f(x,y) = xy$ subject to $2x + 3y = 12$.

D3. ⭐⭐ Find the extrema of $f(x,y) = 3x + 4y$ on the circle $x^2 + y^2 = 25$. (Reproduce the Cauchy–Schwarz pattern of Example 2 in Section 31.8.)

D4. ⭐⭐⭐ Find the point on the plane $x + 2y + 3z = 6$ closest to the origin using Lagrange multipliers (minimize $x^2 + y^2 + z^2$).

D5. ⭐⭐⭐ A rectangular box without a top has volume $32\,\text{m}^3$. Use Lagrange multipliers to find the dimensions minimizing the surface area $S = xy + 2xz + 2yz$ subject to $xyz = 32$.

D6. ⭐⭐⭐ Find the dimensions of the largest-volume rectangular box (faces parallel to the coordinate planes) inscribed in the sphere $x^2 + y^2 + z^2 = 12$. Maximize $V = 8xyz$ for the first-octant corner $(x,y,z)$.

D7. ⭐⭐⭐ Use Lagrange multipliers to find the maximum and minimum of $f(x,y) = x^2 + y^2$ on the ellipse $\frac{x^2}{4} + y^2 = 1$. Interpret geometrically as the longest and shortest "radii."

D8. ⭐⭐ Re-solve the "Check Your Understanding" problem of Section 31.5 — maximize $x + y$ on $x^2 + y^2 \le 2$ — using a single Lagrange equation, and confirm you get $(1,1)$ with value $2$.

Part E — Applications: Economics, Engineering, Data Science (Sections 31.10–31.11)

E1. ⭐⭐⭐ (Economics — utility.) A consumer has utility $U(x,y) = x^{1/2}y^{1/2}$ over goods $x, y$ with prices $p_x = 2$, $p_y = 4$ and budget $B = 40$. Use Lagrange multipliers to find the optimal bundle. Verify the equal-marginal-utility-per-dollar condition $U_x/p_x = U_y/p_y$ from Section 31.10, and state the value of $\lambda$ (the marginal utility of income).

E2. ⭐⭐⭐⭐ (Economics — Cobb–Douglas.) A firm produces $Q = A L^{a}K^{b}$ with wage $w$, capital rental $r$, and cost budget $wL + rK = B$. Derive the optimal input demands $L^* = \frac{aB}{w(a+b)}$ and $K^* = \frac{bB}{r(a+b)}$ from scratch (Section 31.10). Then specialize to $A = 1$, $a = b = \tfrac12$, $w = 1$, $r = 4$, $B = 100$ and give the numerical optimum.

E3. ⭐⭐⭐ (Engineering — material minimization.) A closed cylindrical can must hold $V = 1000\,\text{cm}^3$. Minimize the surface area $S = 2\pi r^2 + 2\pi r h$ subject to $\pi r^2 h = 1000$ (Lagrange multipliers or substitution). Show the optimal can has $h = 2r$.

E4. ⭐⭐⭐⭐ (Data science — least squares.) Fit a line $y = mx + c$ to $(0,1)$, $(1,1)$, $(2,4)$ by minimizing $S(m,c) = \sum_i (y_i - mx_i - c)^2$. Set $\nabla S = \mathbf 0$, solve the resulting normal equations (Section 31.11) by hand, and report $m$ and $c$. Then confirm the Hessian of $S$ is positive-definite, so the critical point is the global minimum (no saddles).

E5. ⭐⭐⭐⭐ (Statistics — constrained MLE.) Observations fall into three categories with counts $n_1, n_2, n_3$ (total $n$) and probabilities $p_1, p_2, p_3$. The log-likelihood is $\ell = n_1\ln p_1 + n_2 \ln p_2 + n_3\ln p_3$ subject to $p_1 + p_2 + p_3 = 1$. Use one Lagrange multiplier to show the MLE is $\hat p_i = n_i/n$, and identify the value of $\lambda$ at the optimum.

E6. ⭐⭐⭐ (Physics — equilibrium.) A particle's potential energy on $x^2 + y^2 = 1$ is $U(x,y) = x + y^2$. Find all equilibrium positions (constrained critical points of $U$) via Lagrange multipliers, and identify the lowest- and highest-energy configurations by comparing values.

Part F — Conceptual and Proof (Sections 31.3–31.12)

F1. ⭐⭐ Explain in your own words why the discriminant test is exactly the "complete the square on the quadratic form" argument of Section 31.4. What does $D > 0$ guarantee about the two squared coefficients?

F2. ⭐⭐⭐ Prove that if $f(x,y)$ is a sum of squares of linear functions plus a constant, every critical point is a global minimum. Connect this to why least squares (Section 31.11) has no saddles.

F3. ⭐⭐⭐ The Lagrange condition $\nabla f = \lambda\nabla g$ is necessary but not sufficient. Give an explicit example where it locates a point that is neither a max nor a min on the constraint, and explain how the "evaluate and compare" step (the Warning in Section 31.8) catches it.

F4. ⭐⭐⭐⭐ (Convexity.) Using the definition in Section 31.12, prove that $f(x,y) = x^2 + y^2$ is convex directly from the chord inequality (no Hessian). Then state why convexity guarantees gradient descent (Chapter 30) finds the global minimum from any start.

Working tip. For every Lagrange answer, run the sanity check from the Warning in Section 31.8: confirm $\nabla g \ne \mathbf 0$ at your point, then evaluate $f$ at all candidates and compare. The multiplier method hands you suspects, not verdicts. For unconstrained problems, never skip the discriminant — and when $D = 0$, fall back to the path-testing method of Example 3 in Section 31.4. Selected answers appear in appendices/answers-to-selected.md.