Study sheet: Multivariable Functions and Optimization

📋 Course Outline

1. Functions of Two Variables
2. Domains of Multivariable Functions
3. Limits and Continuity
4. Partial Derivatives and Differentials
5. Second Partial Derivatives
6. Homogeneous Functions
7. Euler Theorem and Production
8. Jacobian Matrix
9. Unconstrained Extrema
10. Equality-Constrained Optimization

📖 1. Functions of Two Variables

🔑 Key Concepts & Definitions

• Function of Two Variables : assigns a unique real number f(x,y) to each ordered pair (x,y) in its domain D

📝 Essential Points

• An n-tuple (x₁,x₂,…,xₙ) is an ordered set of n elements used as the input of a function of several variables.

📖 2. Domains of Multivariable Functions

🔑 Key Concepts & Definitions

• Function Domain : the subset of ℝ² containing every ordered pair for which the function is defined

📝 Essential Points

• For f₁(x,y)=(-x²+y³-5)/(x-1), the domain is $D_{f_1}=\{(x,y)\in\mathbb{R}^2\mid x\ne1\}$ because the denominator cannot be zero.

📖 3. Limits and Continuity

🔑 Key Concepts & Definitions

• Multivariable Limit : The limit of f(x,y) as (x,y) approaches (x₀,y₀) is the finite number L approached along every possible path to (x₀,y₀).
• Continuity : continuous at (x₀,y₀) when f(x₀,y₀) is defined and $\lim_{(x,y)\to(x_0,y_0)}f(x,y)=f(x_0,y_0)$

📝 Essential Points

📌 If f and g are continuous at (x₀,y₀), then kf, f±g, fg, and f/g are continuous there, provided g(x₀,y₀)≠0 for the quotient.

💡 Memory Hook

A limit concerns approach along every path, whereas continuity also requires the function value to be defined and equal to that limit.

📖 4. Partial Derivatives and Differentials

🔑 Key Concepts & Definitions

• Partial Derivative : The partial derivative of z=f(x,y) with respect to x differentiates with respect to x while treating y as a constant, and the derivative with respect to y treats x as a constant.

★ Must-know

📐 Formula — For z=f(x,y), the total differential is $dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy=f'_x(x,y)dx+f'_y(x,y)dy$.

Further detail

📌 Partial derivatives obey linearity, the product rule, and the reciprocal rule: $\frac{\partial(\alpha f+\beta g)}{\partial x_i}=\alpha\frac{\partial f}{\partial x_i}+\beta\frac{\partial g}{\partial x_i}$, $\frac{\partial(fg)}{\partial x_i}=g\frac{\partial f}{\partial x_i}+f\frac{\partial g}{\partial x_i}$, and $\frac{\partial(1/f)}{\partial x_i}=-\frac{1}{f^2}\frac{\partial f}{\partial x_i}$ when f≠0.

💡 Memory Hook

Freeze one variable while the other moves, then combine both movements in the total differential.

📖 5. Second Partial Derivatives

★ Must-know

📌 If f_xy and f_yx are continuous on ℝ², then $f_{xy}(x,y)=f_{yx}(x,y)$ at every point.

Further detail

• For f(x,y)=5x²−xy+2y² and f(x,y)=x²sin(y), the course presents applications of equality between mixed partial derivatives.

📖 6. Homogeneous Functions

🔑 Key Concepts & Definitions

• Homogeneous Function : A function f is homogeneous of degree k when $f(tx,ty)=t^k f(x,y)$.

📝 Essential Points

📐 Formula — For n variables, a continuously partially differentiable function is homogeneous of degree k if and only if $x_1f_{x_1}(x)+x_2f_{x_2}(x)+\cdots+x_nf_{x_n}(x)=kf(x)$ for all points in its domain under the stated scaling condition.

💡 Memory Hook

Scaling the inputs by t scales the output by t^k.

📖 7. Euler Theorem and Production

🔑 Key Concepts & Definitions

• Cobb-Douglas Function : $Q=f(L,K)=AL^\alpha K^\beta$ with A, α, and β positive, where Q is total production, L is labour, and K is capital

📝 Essential Points

📐 Formula — For the Cobb-Douglas function, scaling both inputs by t gives $f(tL,tK)=t^{\alpha+\beta}f(L,K)$.

• Returns to scale are classified by:
  • α+β=1: constant returns
  • α+β>1: increasing returns
  • α+β<1: decreasing returns

💡 Memory Hook

Scale inputs → identify degree → apply Euler’s equation → classify returns to scale.

📖 8. Jacobian Matrix

🔑 Key Concepts & Definitions

• Jacobian Matrix : collects the first partial derivatives of functions of several variables, and its determinant provides information about their dependence

📝 Essential Points

📌 For the functions y₁=x₁+x₂ and y₂=x₁²+2x₁x₂+x₂², the Jacobian determinant is zero because $|J|=(2x_1+2x_2)-(2x_1+2x_2)=0$, so the functions are dependent and y₂=y₁².

💡 Memory Hook

A zero Jacobian determinant indicates dependence, whereas a nonzero determinant indicates independence.

📖 9. Unconstrained Extrema

🔑 Key Concepts & Definitions

• Global and Local Extrema : A global maximum or minimum compares a function value with every point in its domain, whereas a local maximum or minimum compares it only with nearby points.
• Saddle Point : A saddle point is a point where the function increases in one direction and decreases in another direction.

📝 Essential Points

📌 If f has an extremum at (x₀,y₀) and has first partial derivatives there, then $f_x(x_0,y_0)=f_y(x_0,y_0)=0$ is necessary but not sufficient.

📐 Formula — For $H=\begin{vmatrix}r&s\s&t\end{vmatrix}=rt-s^2$, where r=f_xx(x₀,y₀), s=f_xy(x₀,y₀), and t=f_yy(x₀,y₀), H>0 with r>0 gives a minimum and H>0 with r<0 gives a maximum.

📌 If H<0 the critical point is a saddle point, while if H=0 the second-order test is inconclusive.

💡 Memory Hook

First derivatives zero → Hessian determinant → minimum, maximum, saddle, or inconclusive.

📖 10. Equality-Constrained Optimization

🔑 Key Concepts & Definitions

• Optimization Problem : In an optimization problem, the objective function is the dependent variable to maximize or minimize, while the independent variables are choice or policy variables whose values determine the optimum.
• Lagrangian : For the equality constraint g(x,y)=c, the Lagrangian is $L(x,y,\lambda)=f(x,y)-\lambda(g(x,y)-c)$, where λ is the Lagrange multiplier.

★ Must-know

• The first-order Lagrange conditions are $L_x=0$, $L_y=0$, and $L_\lambda=0$, equivalently $f_x=\lambda g_x$, $f_y=\lambda g_y$, and $g(x,y)=c$.

• The constrained classification is:

  • H>0: maximum
  • H<0: minimum
  • H=0: no extremum

Further detail

• For f(x,y)=x²−y²+xy+5x subject to g(x,y)=x−2y=0, the Lagrange method gives the critical point (x₀,y₀)=(-2,-1) with λ=0 and a bordered Hessian value of -10, so the point is a constrained minimum.

💡 Memory Hook

Build the Lagrangian → set three first-order conditions → classify with the bordered Hessian.

📊 Synthesis Tables

Returns to Scale

Condition	Classification	Output response
α+β=1	Constant returns	Output moves in the same proportions as factors
α+β>1	Increasing returns	Output expands more proportionally than factors
α+β<1	Decreasing returns	Output expands less proportionally than factors

Unconstrained Hessian Test

Condition	Conclusion
H>0 and r>0	Local minimum
H>0 and r<0	Local maximum
H<0	Saddle point
H=0	Inconclusive