Global and Local Extrema Calculus
Page 1: Introduction to Extrema of a Function
Overview of extrema in functions
Topics include:
Local behaviour of functions
Global behaviour of functions
Focus on the local and global behaviour of Taylor’s Polynomials
Page 2: Topics Overview
Topics to be covered in detail:
Extrema of a function
Local behaviour and Taylor’s polynomial
Global behaviour
Page 3: Definition of Extrema
Extrema: Points where a function attains maximum or minimum values.
The function f is continuous in the closed interval [a, b]:
Weierstrass theorem states there exist points where f reaches its maximum and minimum.
Page 4: Classifying Extrema
Local Maximum/Minimum:
Local maximum at x0: Exists δ such that f(x) ≤ f(x0) for all x within |x - x0| < δ.
Local minimum at x0: Exists δ such that f(x) ≥ f(x0) for all x within |x - x0| < δ.
Global Maximum/Minimum:
Global maximum at x0: f(x) ≤ f(x0) for all x in [a, b].
Global minimum at x0: f(x) ≥ f(x0) for all x in [a, b].
Page 5: Properties of Global and Local Extrema
If x0 is a global extremum, it is also a local extremum.
The number of extremum points can be finite or infinite.
Global extrema may be located at endpoints a and/or b.
Page 6: Theorems on Local Behaviour
Theorem 1: If function f is differentiable and has a local extremum at x0, then f'(x0) = 0.
Converse may not hold; points where f' vanishes are critical points, candidates for local extrema.
Theorem 2:
If f is twice differentiable and f'(x0) = 0:
If f''(x0) > 0: local minimum at x0.
If f''(x0) < 0: local maximum at x0.
Page 7: Example of Critical Points
Example function:
f(x) = x^3 - 3x + 1
f'(x) = 3x^2 - 3 = 0 gives critical points at x = ±1.
Second derivative f''(x) = 6x used to classify:
f''(1) > 0 (local minimum at x = 1);
f''(-1) < 0 (local maximum at x = -1).
Page 8: Inflection Points
Convexity and Concavity:
Function f is convex if the segment between any two points in (a, b) lies above the graph of f.
Function f is concave if it lies below the graph.
Inflection Point: Change in curvature from concave to convex or vice versa.
Page 9: Conditions for Convexity/Concavity
Theorem:
If f''(x) > 0 for all x in (a, b), then f is convex in (a, b).
If f''(x) < 0, then f is concave in (a, b).
Inflection points occur at f''(x0) = 0 and indicate local extrema of f'.
Page 10: Taylor's Polynomial and Local Behaviour
Taylor's Polynomial provides information on local behaviour around critical points.
For an infinitely differentiable function at x0 with f’(x0) = 0:
Formulation:
f(x) = f(x0) + (f^(p)(x0)/p!)(x - x0)^p + o(|x - x0|^p), where p is the order of the first non-zero derivative.
Page 11: Conditions Based on Derivative Order
Conditions based on the order p of the first derivative at x0:
If p is even and f^(p)(x0) > 0: local minimum at x0 (convex).
If p is even and f^(p)(x0) < 0: local maximum at x0 (concave).
If p is odd and f^(p)(x0) > 0: inflection point at x0 (strictly increasing).
If p is odd and f^(p)(x0) < 0: inflection point at x0 (strictly decreasing).
Page 12: Examples Using Taylor's Polynomial
Example 1: f(x) = x^7 sin^4(x), with Maclaurin polynomial of degree 11 indicates inflection point at x = 0.
Example 2: f(x) = cos^3(x) log2(1 + x), with degree 2 polynomial shows a local minimum at x = 0 and convexity there.
Page 13: Steps to Determine Global Extrema
Calculate critical points of f in (a, b).
Identify points where f is not differentiable.
Evaluate at endpoints x = a and x = b.
Compare function values from steps 1-3 to find global extrema.
Points with the highest and lowest values indicate global maximum and minimum.
Page 14: Global Extrema Example
Consider function: f(x) = 2x^(5/3) + 5x^(2/3) in the interval [-2, 1].
Finding critical points and evaluating boundaries:
f'(-1) = 3, f(0) = 0, f(-2) = 41/3, f(1) = 7.
Result: Global minimum at x = 0, and global maximum at x = 1.