Pattern 1: Monotonicity with Parameters (Most Common) "Find values of k for which f(x) = $x^3$ + $kx^2$ + 5x + 1 is increasing for all x." Approach: f'(x) = 3$x^2$ + 2kx + 5 >= 0 for all x. Since the leading coefficient is positive, require discriminant <= 0: 4$k^2$ - 60 <= 0, giving -sqrt(15) <= k <= sqrt(15).

This pattern appears almost every year with variations: decreasing on an interval, increasing on a specific domain, or monotonicity of composite functions.

Pattern 2: Number of Extrema "Find the number of local maxima and minima of f(x) = x^$\frac{2}{3}$(6 - x)^$\frac{1}{3}$." Approach: Find f'(x) using the product rule, identify critical points (including where f' is undefined due to fractional exponents), then use sign analysis. These problems test careful algebraic manipulation.

Pattern 3: Global Max/Min on Closed Interval "Find the maximum and minimum values of f(x) = sin(x) + cos(x) on [0, 2pi]." Approach: f'(x) = cos(x) - sin(x) = 0 gives x = pi/4, 5pi/4. Evaluate f at x = 0, pi/4, 5pi/4, 2pi. Maximum = sqrt(2) at x = pi/4, minimum = -sqrt(2) at x = 5*pi/4.

Pattern 4: MVT Verification "Verify MVT for f(x) = x(x - 1)(x - 2) on [0, 1/2] and find c." Approach: Verify conditions (polynomial, so automatic). Compute slope, then solve f'(c) = slope. These are direct computation problems.

Pattern 5: Optimization "A wire of length 28 cm is bent into a rectangle. Find dimensions for maximum area." Approach: 2(l + b) = 28, maximize A = l * b = l(14 - l). Critical point: l = 7, so it's a square. Maximum area = 49 $cm^2$.

Pattern 6: Tangent/Normal "Find the equation of the tangent to y = $x^3$ - 3x + 2 at the point where x = 1." Approach: y(1) = 0, y'(1) = 0. Tangent: y = 0 (horizontal tangent). Normal: x = 1 (vertical line).

Pattern 7: Proving Inequalities Using Monotonicity "Prove that x - $x^3$/6 < sin(x) < x for x > 0." Approach: Define g(x) = x - sin(x), show g'(x) = 1 - cos(x) >= 0, so g is increasing. Since g(0) = 0, g(x) > 0 for x > 0, proving sin(x) < x. Similarly for the lower bound.

Frequency Analysis:

• Monotonicity with parameters: ~30% of questions
• Optimization: ~25%
• Maxima/minima finding: ~20%
• MVT/Rolle's: ~10%
• Tangent/normal: ~10%
• Proving inequalities: ~5%