Understanding Stationary Points and Their Nature

May 1, 2025

Chord Mask Video - Stage 3 Points

Overview

  • Two main objectives:
    1. Use differentiation to find the coordinates of stationary points.
    2. Use the second derivative to determine whether a stationary point is a maximum or minimum.
  • Key Concepts:
    • Stationary points, also known as turning points, are where the graph turns.
    • Maximum points (maxima) have a positive gradient that turns negative.
    • Minimum points (minima) have a negative gradient that turns positive.
    • The gradient (dy/dx) is zero at stationary points.
    • The second derivative helps determine the nature of turning points.
      • Less than zero for maxima.
      • Greater than zero for minima.

Finding Coordinates of Stationary Points

Example 1: Quadratic Function

  • Function: y = x² - 6x + 4
  • Process:
    1. Differentiate to find gradient function: dy/dx = 2x - 6
    2. Set dy/dx = 0 to find stationary point:
      • 2x - 6 = 0
      • Solve for x: x = 3
    3. Substitute x = 3 back into the original equation to find y:
      • y = 3² - 18 + 4 = -5
    4. Result: Minimum point at (3, -5)

Example 2: Cubic Function

  • Function: y = x³ - 3x² - 24x
  • Process:
    1. Differentiate to find gradient function: dy/dx = 3x² - 6x - 24
    2. Set dy/dx = 0:
      • 0 = 3x² - 6x - 24
      • Divide by 3: 0 = x² - 2x - 8
      • Factorize: (x-4)(x+2) = 0
      • Solve: x = 4, x = -2
    3. Substitute x-values back into original equation for y:
      • x = 4: y = -80
      • x = -2: y = 28
    4. Results: Maximum point at (-2, 28), Minimum point at (4, -80)

Determining Nature of Stationary Points

Second Derivative Test

  • For maxima: d²y/dx² < 0
  • For minima: d²y/dx² > 0

Example: Using Second Derivative

  • Initial Example: Maximum (-2, 28), Minimum (4, -80)
  • Function: d²y/dx² = 6x - 6
    • Substitute x = -2:
      • d²y/dx² = -18 (negative, confirms maximum)
    • Substitute x = 4:
      • d²y/dx² = 18 (positive, confirms minimum)

Typical Question: Find Coordinates and Nature

  • Function: y = x - x² - x³
  • Process:
    1. Differentiate: dy/dx = 1 - 2x - 3x²
    2. Set dy/dx = 0:
      • Solve: 3x² + 2x - 1 = 0
      • Factorize and solve: x = 1/3, x = -1
    3. Substitute x-values for y-values:
      • x = -1: y = -1
      • x = 1/3: y = 5/27
    4. Determine nature using second derivative:
      • d²y/dx² = -2 - 6x
      • x = -1: d²y/dx² = 4 (minimum)
      • x = 1/3: d²y/dx² = -4 (maximum)
    5. Results: Minimum at (-1, -1), Maximum at (1/3, 5/27)

Summary

  • Use dy/dx = 0 to find stationary points.
  • Use d²y/dx² to find the nature (maxima or minima).
  • Maxima: d²y/dx² < 0
  • Minima: d²y/dx² > 0