Lesson 6-2: Derivatives – Extrema and Optimization

Clinic 1: Box Volume Maximization

• Objective: V = x(20-2x)(15-2x) from cut corners, domain 0 < x < 7.5.
• V'(x) = 12x² - 140x + 300, critical points: x = 3.33, x = 7.5.
• Test: V''(3.33) < 0 (max), check endpoints V(0) = V(7.5) = 0.
• Maximum: V(3.33) ≈ 592.6 cubic units, feasible for practical box.

Clinic 2: Fence Perimeter Problem

• Objective: A = xy with 2x + y = 240 constraint, domain x > 0.
• Substitute: A(x) = x(240-2x) = 240x - 2x², A'(x) = 240 - 4x.
• Critical point: x = 60, A''(x) = -4 < 0 confirms maximum.
• Optimal: 60×120 rectangle, area = 7200 sq units within budget.

Clinic 3: Can Surface Area

• Objective: S = 2πr² + 2πrh with V = πr²h = 355 constraint.
• Substitute h = 355/(πr²): S(r) = 2πr² + 710/r, S'(r) = 4πr - 710/r².
• Critical point: r³ = 710/(4π), r ≈ 3.84 cm, S''(r) > 0 (min).
• Minimum surface: r = 3.84 cm, h = 7.68 cm, real manufacturing size.

Clinic 4: Distance Minimization

• Objective: d = √[(x-2)² + (x²-1)²] from point (2,1) to parabola y = x².
• Minimize d²: f(x) = (x-2)² + (x²-1)², f'(x) = 2(x-2) + 4x(x²-1).
• Solve f'(x) = 0: 2x - 4 + 4x³ - 4x = 0 → 4x³ - 2x - 4 = 0.
• Numerical solution: x ≈ 1.22, minimum distance ≈ 0.86 units.

Clinic 5: Profit Maximization

• Revenue R(x) = 50x - 0.1x², Cost C(x) = 10x + 1000, domain x ≥ 0.
• Profit P(x) = R(x) - C(x) = 40x - 0.1x² - 1000, P'(x) = 40 - 0.2x.
• Critical point: x = 200 units, P''(x) = -0.2 < 0 confirms max.
• Maximum profit: P(200) = $3000 at production level = 200 units.

Clinic 6: Ladder Around Corner

• Objective: L = a/sin θ + b/cos θ where a,b are corridor widths.
• L'(θ) = -a cos θ/sin² θ + b sin θ/cos² θ, set equal to zero.
• Critical condition: (a/b) = (sin³ θ)/(cos³ θ) = tan³ θ.
• Minimum ladder length depends on specific corridor dimensions.

Clinic 7: Inventory Cost Model

• Cost C(x) = (D/x)×S + (x/2)×H where D=demand, S=setup, H=holding.
• C'(x) = -DS/x² + H/2, critical point: x = √(2DS/H).
• EOQ formula: x* = √(2DS/H), C''(x*) > 0 confirms minimum.
• Optimal order quantity minimizes total inventory costs.

Clinic 8: Pipeline Route

• Cost = land portion + underwater portion with different unit costs.
• Total C(x) = c₁√(a² + x²) + c₂√(b² + (d-x)²) where x is split point.
• C'(x) = c₁x/√(a² + x²) - c₂(d-x)/√(b² + (d-x)²) = 0.
• Solve numerically: optimal split minimizes construction cost.

Clinic 9: Window Design

• Perimeter P = 2x + y + πx/2, Area A = xy + πx²/8, P fixed.
• Express y = P - 2x - πx/2, substitute into A(x).
• A'(x) = P - 4x - πx = 0, critical point x = P/(4+π).
• Maximum area window has optimal rectangle-semicircle ratio.

Clinic 10: Reflector Design

• Light path: source to reflector point to target, minimize total distance.
• Use Fermat's principle: angle of incidence = angle of reflection.
• Set up distance function and differentiate to find reflection point.
• Optimal design achieves minimum travel time for light ray.

Concavity and Extrema

If $f'(c)=0$ and $f''$ exists near c: $f''(c)<0$ implies a local maximum; $f''(c)>0$ implies a local minimum; $f''(c)=0$ is inconclusive.

Inflection Points

Points where concavity changes sign are inflection points. Often found by solving $f''(x)=0$ or where$f''$ is undefined and confirming sign change.

Fence Problem (No Roof)

Perimeter 20 m, three sides of a rectangle (one side along a wall). Maximize area $S=x(20-2x)$ for$0<x<10$. Critical $x=5$ → maximum area 50 m².

Box with Square Base (No Lid)

Surface area 108 m², base $x imes x$, height h. Constraint $x^2+4xh=108$ →$h=(108-x^2)/(4x)$; volume $V=x^2h=27x- frac{1}{4}x^3$ maximized at x=6, h=3, V=108.

Polynomial Mix

1. For f(x)=x^4-4x^2+1, find intervals of increase/decrease.
2. Classify critical points and identify inflection points.
3. Sketch a qualitative graph with extrema and concavity.

Rational Function

1. For g(x)=(x+1)/(x^2+1), locate extrema and concavity.
2. Determine asymptotic behavior and discuss global extrema.
3. Explain why no vertical asymptotes occur.

Exponential-Log

1. For $h(x)=xe^{-x}$, find absolute max on $[0,\infty)$.
2. For $p(x)=x\ln x$ ($x>0$), locate critical points and nature.
3. Compare growth/decay impact on extrema.

Trigonometric

1. For q(x)=sin x - 1/2 sin 2x on [0,2π], find extrema.
2. Discuss concavity intervals using q''(x).
3. Interpret geometric meaning on the unit circle.

Packaging Design

Minimize surface area of a closed cylinder of fixed volume V. Derive r and h for minimal material and discuss manufacturability.

1. Express h in terms of r using volume constraint.
2. Differentiate surface area S(r) and find the minimizer.
3. Conclude r:h ratio and practical implications.

Logistics Path

Minimize travel time across road (speed v1) and field (speed v2) between two points separated by a river bank.

1. Model time as function of landing point x; set domain.
2. Compute derivative and solve for optimal x.
3. Discuss sensitivity when v2→v1 and v2≪v1.

Fermat’s Theorem

If f has a local extremum at c and is differentiable at c, then f'(c)=0.

Sketch: Compare one-sided difference quotients; signs must be ≥0 and ≤0.

Second Derivative Test

If f'(c)=0 and f''(c)≠0, the sign of f''(c) determines local min/max.

Sketch: Use Taylor expansion to second order near c.

Case A: Quartic with Two Wells

1. Find f'(x) and solve for candidates
2. Use sign chart to classify local extrema
3. Check inflection via f''(x)
4. Summarize with a qualitative sketch

Case B: Rational with Hole

1. Determine domain and removable discontinuity
2. Locate critical points excluding the hole
3. Classify using first/second derivative tests
4. State limits near the hole and behavior at infinity

Case C: Exponential-Log Mix

1. Differentiate and solve f'(x)=0 numerically
2. Discuss existence/uniqueness of the maximizer
3. Check concavity to confirm classification
4. Report extremum value with units

Bank A: First Derivative Test

1. Build sign chart around candidates
2. Identify increase/decrease intervals
3. State local extrema with coordinates

Bank B: Second Derivative Test

1. Compute f'' at each candidate
2. Classify min/max/undetermined
3. Handle f''=0 via monotonicity

Bank C: Closed Interval

1. Evaluate endpoints
2. Compare with interior candidates
3. Report absolute extrema with units

Bank D: Modeling

1. Form objective+constraints from context
2. Reduce to single variable
3. Solve and justify feasibility

Project A: Cost with Volume Constraint

• Form cost function from material prices
• Use volume constraint to eliminate a variable
• Minimize and interpret sensitivity

Project B: Travel Time Minimization

• Model time with different media speeds
• Differentiate and solve for optimal split
• Discuss limiting cases v2→v1 and v2≪v1

Project C: Revenue Management

• Price–demand model R(x)=x·p(x)
• Find optimal quantity and price
• Check concavity and units

Proof & Theory

• State and explain Fermat’s Theorem on stationary points.
• Relate Rolle’s Theorem and MVT to optimization logic.
• Discuss when second derivative test fails and alternatives.

Applications

• Revenue management with price elasticity assumptions.
• Material minimization under strength constraints.
• Travel-time minimization with piecewise speeds.