Chapter 6 — Application of Derivatives

NCERT Class 12 Maths Chapter 6 covers the Application of Derivatives, teaching students how to use derivatives to find rates of change, determine increasing or decreasing intervals of functions, and locate maximum and minimum values using the First and Second Derivative Tests.

Chapter 6 of NCERT Class 12 Mathematics (Part I) focuses on the Application of Derivatives across engineering, science, and economics. It begins with rates of change using the Chain Rule, then introduces conditions for a function to be increasing or decreasing on an interval using the sign of f'(x). The chapter defines critical points, local maxima and minima, and presents two key tests: the First Derivative Test (sign change of f') and the Second Derivative Test (sign of f''). It concludes with a working rule to find absolute maximum and minimum values on a closed interval by evaluating f at all critical points and endpoints.

1. 01The derivative dy/dx represents the rate of change of y with respect to x; the Chain Rule connects rates of change through an intermediate variable t.
2. 02A function f is increasing on (a, b) if f'(x) > 0 for each x in (a, b), and decreasing if f'(x) < 0 for each x in (a, b).
3. 03A critical point is any point c where f'(c) = 0 or f is not differentiable; critical points are candidates for local maxima or minima.
4. 04First Derivative Test: if f'(x) changes from positive to negative at c, then c is a local maximum; if it changes from negative to positive, c is a local minimum; no sign change means c is a point of inflexion.
5. 05Second Derivative Test: if f'(c) = 0 and f''(c) < 0, then c is a local maximum; if f''(c) > 0, it is a local minimum; if f''(c) = 0, the test fails and the First Derivative Test must be used.
6. 06To find absolute maximum and minimum on a closed interval [a, b], evaluate f at all critical points inside the interval and at the endpoints, then compare all values.