Application of Derivatives: Monotonicity & Maxima-Minima

Application of Derivatives connects the sign and value of derivatives to the behavior of functions — whether they increase, decrease, or achieve extreme values. This topic is one of the highest-weighted in JEE Main calculus.

Monotonicity (Increasing/Decreasing Functions): A function f is strictly increasing on interval I if for all x1 < x2 in I, f(x1) < f(x2). It is strictly decreasing if f(x1) > f(x2). The first derivative provides the test:

• If f'(x) > 0 for all x in (a, b), then f is strictly increasing on [a, b]
• If f'(x) < 0 for all x in (a, b), then f is strictly decreasing on [a, b]
• If f'(x) = 0 for all x in (a, b), then f is constant on [a, b]

Critical Points: A point x = c is a critical point of f if f'(c) = 0 or f'(c) does not exist (but f(c) is defined). Critical points are candidates for local extrema.

First Derivative Test for Local Extrema: At a critical point x = c:

• If f' changes from positive to negative (+ to -), then f has a local maximum at c
• If f' changes from negative to positive (- to +), then f has a local minimum at c
• If f' does not change sign, then c is a point of inflection (no extremum)

Second Derivative Test: At a critical point x = c where f'(c) = 0:

• If f''(c) < 0, then f has a local maximum at c
• If f''(c) > 0, then f has a local minimum at c
• If f''(c) = 0, the test is inconclusive — use the first derivative test or higher derivative test

Global (Absolute) Extrema on [a, b]: For a continuous function on a closed interval [a, b]:

1. Find all critical points in (a, b)
2. Evaluate f at all critical points and at endpoints a and b
3. The largest value is the global maximum; the smallest is the global minimum

Rolle's Theorem: If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c in (a, b) such that f'(c) = 0.

Mean Value Theorem (MVT): If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that f'(c) = [f(b) - f(a)]/(b - a).

Concavity and Points of Inflection:

• f''(x) > 0: concave upward (cup shape)
• f''(x) < 0: concave downward (cap shape)
• Point of inflection: where f'' changes sign (concavity reversal)

Tangent and Normal Lines:

• Tangent at (x0, y0): y - y0 = f'(x0)(x - x0)
• Normal at (x0, y0): y - y0 = -1/f'(x0) * (x - x0) [perpendicular to tangent]

The key problem-solving concept is using the sign chart of f'(x) to determine intervals of increase/decrease and locate extrema, combined with endpoint analysis for global extrema on closed intervals.