Derivatives: Monotonicity and Extrema

Overview

Lecture 3 in the Application of Derivatives series covers increasing and decreasing functions and maxima–minima. Focus: first- and second-derivative tests, interval notation, and exam-oriented problem solving.

Increasing and Decreasing Functions

• A function f(x) is increasing on an interval if f′(x) ≥ 0 there; strictly increasing if f′(x) > 0.
• A function is decreasing on an interval if f′(x) ≤ 0; strictly decreasing if f′(x) < 0.
• Sign of derivative determines monotonicity: f′(x) > 0 ⇒ increasing; f′(x) < 0 ⇒ decreasing.
• For inequalities, “greater than zero” means x in positive region; “less than zero” means negative region.
• For interval notation: use round brackets when endpoints are not included; square when equality holds.
• Infinity never takes a closed bracket; always use (−∞, ...) or (..., ∞).

Factor-Sign Method for Intervals

• Compute f′(x); factor it: f′(x) = k(x − a)(x − b)...
• For increasing (f′(x) ≥ 0): x lies outside roots if factors change sign with a single quadratic product.
• For decreasing (f′(x) ≤ 0): x lies between roots for single quadratic product with positive leading k.
• Quick rule used:
  • If f′(x) > 0: x < smaller root or x > larger root (for product of two linear factors with positive k).
  • If f′(x) < 0: smaller root < x < larger root (strictly decreasing, use open brackets).

Maxima and Minima: Second Derivative Test

• Steps:
  • Compute f′(x), set f′(x) = 0 to get critical points.
  • Compute f″(x).
  • Evaluate f″ at critical points:
    • f″(c) > 0 ⇒ local minimum at x = c (minima).
    • f″(c) < 0 ⇒ local maximum at x = c (maxima).
• Value vs point:
  • “Maxima/minima at x = c” refers to the point c.
  • “Maximum/minimum value” means compute f(c).

Key Worked Patterns

• Simplify f′(x) by factoring and reducing common factors before sign analysis.
• When strict language (“strictly increasing/decreasing”) appears, do not include equality in intervals.
• Bracket care in answers: equality ⇒ square bracket; strict ⇒ round bracket.

Typical Questions and Approaches

• Determine intervals of increase/decrease:
  • Find f′(x); factor; locate roots; apply sign logic; write union of intervals.
• Classify extrema and compute values:
  • Find critical points via f′(x) = 0; test with f″(x); compute f(c) for requested value.
• Distinguish “maximum value” vs “value of x at which maximum occurs” carefully.

Examples: Exam-Style Techniques

• Polynomial example: f(x) with f′(x) factored; increasing where f′(x) ≥ 0 gives x in (−∞, smaller root] ∪ [larger root, ∞) if non-strict.
• Decreasing interval: for f′(x) ≤ 0 strictly, answer is (smaller root, larger root).
• Rational/defined-domain note: if x ≠ 0 is given, ensure domain respects exclusions when classifying monotonicity.

Optimization with Constraint

• Given relation to eliminate variables (e.g., x + y = 100):
  • Express one variable in terms of the other; form single-variable f(x).
  • Differentiate, set f′(x) = 0; test with f″.
  • Example: minimize x² + y² with x + y = 100 ⇒ y = 100 − x; f(x) = x² + (100 − x)²; f′ = 4x − 200 = 0 ⇒ x = 50, y = 50; f″ = 4 > 0 ⇒ minimum.

Logarithmic Example Pattern

• For f(x) = x − ln x (x > 0): f′(x) = 1 − 1/x; set 0 ⇒ x = 1; f″(x) = 1/x² > 0 ⇒ minimum at x = 1; minimum value f(1) = 1 − 0 = 1.
• General handling: convert 1/e to e^(−1) for evaluating logs; use ln(e^(−1)) = −1.

CET/Board Exam Tips

• CET and board both ask MCQs; increasing/decreasing and maxima/minima are commonly tested.
• Read questions precisely: they may ask for x at which maximum occurs, or the maximum value itself.
• Avoid algebraic slips with brackets and strictness; confirm domain restrictions.

Key Terms & Definitions

• Increasing function: f′(x) ≥ 0 on an interval.
• Strictly increasing: f′(x) > 0 on an interval.
• Decreasing function: f′(x) ≤ 0 on an interval.
• Strictly decreasing: f′(x) < 0 on an interval.
• Critical point: value c where f′(c) = 0 (or undefined within domain).
• Maxima (local maximum): f″(c) < 0 at a critical point.
• Minima (local minimum): f″(c) > 0 at a critical point.

Summary Table: Tests and Outcomes

Action Items / Next Steps

• Practice factoring f′(x) and interval sign charts quickly.
• Drill second derivative tests; memorize: f″ > 0 ⇒ minima, f″ < 0 ⇒ maxima.
• Carefully parse questions for “value” vs “point”; match brackets to strictness.
• Review domain restrictions before concluding monotonicity or extrema.](streamdown:incomplete-link)