Step-by-Step Optimization Method:

Step 1: Understand and Draw Read the problem carefully. Draw a diagram if geometric. Identify what quantity must be maximized or minimized (the objective function) and what constraints exist.

Step 2: Assign Variables Let the unknown quantities be x, y, etc. Express the objective function in terms of these variables.

Step 3: Reduce to One Variable Use the constraint equation(s) to eliminate all but one variable. The objective function should be f(x) with a single variable x and a well-defined domain.

Step 4: Find Critical Points Compute f'(x), set f'(x) = 0, and solve. Also check where f'(x) is undefined within the domain.

Step 5: Classify and Verify Use the second derivative test: f''(c) > 0 means minimum, f''(c) < 0 means maximum. Alternatively, check endpoints and compare all values.

Step 6: Answer the Question Substitute back to find the required quantity (which may be different from x).

Standard JEE Optimization Problems:

Geometric optimization:

• Maximum area of rectangle inscribed in a circle/ellipse/triangle/parabola
• Minimum surface area for fixed volume (cylinder, cone, box)
• Maximum volume for fixed surface area
• Shortest path problems (reflection principle)

Algebraic optimization:

• Maximize/minimize expressions like $a^m$ * $b^n$ subject to a + b = constant (use AM-GM or calculus)
• Find the rectangle of maximum area under a given curve

Applied optimization:

• Minimum cost, maximum profit problems
• Optimal angle for maximum range (projectile)
• Closest point on a curve to a given point

Key Results to Remember:

1. Rectangle in circle: Maximum area rectangle inscribed in a circle of radius r is a square with side r*sqrt(2), area = 2$r^2$.

2. Cylinder in sphere: Maximum volume cylinder inscribed in sphere of radius R has h = 2R/sqrt(3), r = R*sqrt$\frac{2}{3}$, volume = $\frac{4*pi*R^3}{(3*sqrt(3)}$).

3. Open box from square sheet: Cut squares of side x = a/6 from each corner of a square sheet of side a. Maximum volume = 2$a^3$/27.

4. Cone of maximum volume from sector: Semi-vertical angle = arcsin$\frac{2}{3}$ for maximum volume cone that can be formed from a circular sector.

5. AM-GM shortcut: If the product of positive terms is constant, their sum is minimized when all terms are equal. If the sum is constant, the product is maximized when all terms are equal.

Common Mistakes:

• Forgetting to check the domain of x (physical constraints)
• Finding a critical point outside the feasible region
• Minimizing when asked to maximize (or vice versa)
• Not verifying that the critical point is indeed a max/min
• Errors in expressing the constraint as a single-variable function