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Understanding Stationary Points and Their Nature
May 1, 2025
Chord Mask Video - Stage 3 Points
Overview
Two main objectives:
Use differentiation to find the coordinates of stationary points.
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:
Differentiate to find gradient function: dy/dx = 2x - 6
Set dy/dx = 0 to find stationary point:
2x - 6 = 0
Solve for x: x = 3
Substitute x = 3 back into the original equation to find y:
y = 3² - 18 + 4 = -5
Result:
Minimum point at (3, -5)
Example 2: Cubic Function
Function:
y = x³ - 3x² - 24x
Process:
Differentiate to find gradient function: dy/dx = 3x² - 6x - 24
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
Substitute x-values back into original equation for y:
x = 4: y = -80
x = -2: y = 28
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/dx ² = -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:
Differentiate: dy/dx = 1 - 2x - 3x²
Set dy/dx = 0:
Solve: 3x² + 2x - 1 = 0
Factorize and solve: x = 1/3, x = -1
Substitute x-values for y-values:
x = -1: y = -1
x = 1/3: y = 5/27
Determine nature using second derivative:
d²y/dx² = -2 - 6x
x = -1: d²y/dx² = 4 (minimum)
x = 1/3: d²y/dx² = -4 (maximum)
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
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