Application of Derivatives | Class 12 Mathematics Notes

Increasing and Decreasing Functions

This section explains how the derivative of a function helps determine whether the function is increasing or decreasing over an interval. A function f(x) is said to be increasing on an interval if for any two points x1 and x2 in the interval, with x1 < x2, we have f(x1) ≤ f(x2). Similarly, it is decreasing if f(x1) ≥ f(x2). The derivative f'(x) provides a test for this behavior: if f'(x) > 0 for all x in an interval, then f is increasing there; if f'(x) < 0, then f is decreasing. The section also discusses the concept of stationary points where f'(x) = 0 and how these points can be candidates for local maxima or minima. The section includes examples showing how to determine intervals of increase and decrease by analyzing the sign of the derivative and interpreting the results graphically. This understanding is crucial for sketching graphs and analyzing function behavior in calculus.

📊 Diagram: Graph showing a curve with intervals marked where derivative is positive (increasing) and negative (decreasing), and points where derivative is zero (stationary points).

🧪 Activity: Activity includes finding intervals of increase and decrease for given functions and verifying with graphs.

🔗 Connection: Leads to the next section on 'Tangents and Normals' by applying derivatives to geometric problems.