Unit 2: Maxima and Minima

1. Maxima and Minima of Functions of One Variable

These are the "peaks" (local maxima) and "valleys" (local minima) of a function y = f(x).

Step-by-Step Procedure:

1. Find Critical Points: Find f'(x) and set it to zero: f'(x) = 0. Solve for x. These solutions are the critical points (or stationary points).
2. Apply Second Derivative Test: Find the second derivative, f''(x).
   • If f''(c) < 0: The function is concave down. 'c' is a Local Maximum.
   • If f''(c) > 0: The function is concave up. 'c' is a Local Minimum.
   • If f''(c) = 0: The test fails. You must use the First Derivative Test (check the sign of f'(x) *around* 'c'. If it changes from + to -, it's a max. If from - to +, it's a min).

2. Maxima and Minima of Functions of Two Variables

For a function z = f(x, y), we are looking for "hills" (maxima), "bowls" (minima), or "passes" (saddle points).

Step-by-Step Procedure:

1. Find Critical Points:
   Find the first-order partial derivatives and set them *both* to zero.
   ∂f/∂x = 0
   ∂f/∂y = 0
   Solve this system of equations to find the critical points (a, b).
2. Calculate Second-Order Derivatives:
   At each critical point (a, b), calculate the following:
   r = ∂²f/∂x²
   s = ∂²f/∂x∂y (the mixed partial)
   t = ∂²f/∂y²
3. Apply the Test (Discriminant):
   Calculate the Discriminant D = rt - s².
   • If D > 0 and r < 0: The point (a, b) is a Local Maximum.
   • If D > 0 and r > 0: The point (a, b) is a Local Minimum.
   • If D < 0: The point (a, b) is a Saddle Point (neither max nor min).
   • If D = 0: The test is inconclusive. Further investigation is needed.

3. Constrained Optimization (Lagrange's Multipliers)

This method is used to find the maximum or minimum of a function f(x, y) subject to a constraint g(x, y) = k.

Example: Find the maximum area of a rectangle (function f) given a fixed perimeter (constraint g).

Step-by-Step Procedure:

1. Identify functions:
   • Objective Function: f(x, y) (the one to be maximized/minimized).
   • Constraint Function: g(x, y) = k.
2. Form the Lagrangian Function (L):
   This new function introduces a new variable, λ (lambda), the Lagrange multiplier.
   L(x, y, λ) = f(x, y) - λ · (g(x, y) - k)
3. Find Partial Derivatives:
   Find the partial derivatives of L with respect to *all* its variables (x, y, and λ) and set them all to zero.
   ∂L/∂x = 0 => ∂f/∂x - λ(∂g/∂x) = 0
   ∂L/∂y = 0 => ∂f/∂y - λ(∂g/∂y) = 0
   ∂L/∂λ = 0 => -(g(x, y) - k) = 0 => g(x, y) = k
4. Solve the System:
   You now have a system of three equations and three unknowns (x, y, λ). Solve them to find the candidate points (x, y).
5. Test the Points:
   Evaluate f(x, y) at each candidate point. The largest value is the constrained maximum, and the smallest is the constrained minimum.

4. Tracing of Curves

A systematic process to sketch the graph of a function.

1. Cartesian Form (y = f(x) or f(x, y) = 0)

1. Symmetry:
   • About Y-axis: If f(-x, y) = f(x, y) (all powers of x are even).
   • About X-axis: If f(x, -y) = f(x, y) (all powers of y are even).
   • About Origin: If f(-x, -y) = f(x, y).
2. Origin: Check if (0, 0) satisfies the equation. If yes, find the tangent(s) at the origin by equating the lowest degree term to zero.
3. Intercepts: Find x-intercepts (set y=0) and y-intercepts (set x=0).
4. Asymptotes: Find any horizontal, vertical, or oblique asymptotes (see next section).
5. Critical Points: Find dy/dx, set to 0 for local max/min.
6. Points of Inflexion: Find d²y/dx², set to 0 (see section 6).
7. Region: Check for values of x or y that make the function undefined (e.g., negative inside a square root, or division by zero).

2. Polar Form (r = f(θ))

1. Symmetry:
   • About Initial Line (x-axis): If f(-θ) = f(θ).
   • About Pole (origin): If f(θ+π) = f(θ) or r is replaced by -r.
   • About Line θ=π/2 (y-axis): If f(π - θ) = f(θ).
2. Pole: Check if r=0 for any value of θ. This gives tangents at the pole.
3. Table of Values: Check key values of θ (0, π/6, π/4, π/3, π/2, etc.) and find r.
4. Asymptotes: More complex, but can exist.

3. Parametric Form (x = f(t), y = g(t))

1. Symmetry: Check for t and -t.
2. Intercepts: Find t for x=0 (y-intercept) and t for y=0 (x-intercept).
3. Asymptotes: Check t-values where x→∞ or y→∞.
4. Tangent: Find dy/dx = (dy/dt) / (dx/dt). Find where it's 0 (horizontal) or ∞ (vertical).
5. Region: Find the range of x and y based on the domain of t.

5. Asymptotes

An asymptote is a line that a curve approaches as it heads towards infinity.

1. Vertical Asymptotes

Occur where the function value goes to ±∞. Typically, this is where the denominator of a rational function is zero.

Rule: If lim (x→a) f(x) = ±∞, then x = a is a vertical asymptote.

2. Horizontal Asymptotes

Occur if the function approaches a finite value 'L' as x goes to ±∞.

Rule: If lim (x→∞) f(x) = L or lim (x→-∞) f(x) = L, then y = L is a horizontal asymptote.

3. Oblique (Slant) Asymptotes

Occur when the degree of the numerator is exactly one more than the degree of the denominator. The asymptote is a line y = mx + c.

How to find:

1. Find m = lim (x→∞) [ f(x) / x ]
2. Find c = lim (x→∞) [ f(x) - mx ]

6. Flexes, Concavity, Convexity, and Points of Inflexion

Concavity and Convexity

This describes the "curvature" of the function.

• Concave Up (or Convex): The graph is "holding water" (like a U-shape).
  • Condition: f''(x) > 0
• Concave Down (or Concave): The graph is "spilling water" (like an ∩-shape).
  • Condition: f''(x) < 0

Points of Inflexion

A point of inflexion is a point on a curve where the concavity changes (from up to down, or from down to up).

How to find:

7. Singular Points

A singular point is a point on a curve f(x, y) = 0 where the curve behaves unusually. These occur where *both* partial derivatives are zero simultaneously:

To classify a singular point, we use the Discriminant D = s² - rt (note this is the reverse of the D for maxima/minima, D = rt - s²).

Let r = ∂²f/∂x², s = ∂²f/∂x∂y, t = ∂²f/∂y² at the point.

• If s² - rt < 0: It is an Isolated Point (the point satisfies the equation, but no other nearby points do).
• If s² - rt > 0: It is a Node (a self-intersection, where two real, distinct tangents exist).
• If s² - rt = 0: It is (usually) a Cusp (a sharp point where two tangents are coincident).