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Contents
First Derivative Test
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First
Derivative Test
The
first derivative test is used to determine the nature of stationary points in
an equation. The second method, the second derivative test, is explained on
the next page.
Given
the equation y = f(x) has a stationary point at (n, f(n)). To determine its
nature, check the sign of f'(x) when x = n- and x = n+
(i.e. the derivative when x is slightly smaller and slightly larger than n).
If the point is a maximum turning point, then
f'(n-) > 0 and f'(n+) < 0
If the point is a minimum turning point, then
f'(n-) < 0 and f'(n+) > 0
If the point is a stationary point of inflexion, then the sign of both f'(n-)
and f'(n+) are the same, either both less than or both more
than zero.
Example:
 Find
the coordinates of the stationary points on the curve x4 + y4
-2x2y2 = 25.
Solution:
Differentiating
w.r.t. x,
At
stationary points, dy/dx = 0.
The
coordinates are
 .
Example:
Find
the coordinates and determine the nature of the stationary points of the
equation y = x3 - 7x2 + 11x - 5.
Solution:
dy/dx = 3x2 - 14x + 11
= (3x - 11)(x - 1)
At
stationary points, dy/dx = 0
(3x - 11)(x - 1) = 0
x = 11/3 or x = 1
When
x = 11/3, y = -256/27,
(11/3,
-256/27) is a minimum turning point.
When
x = 1, y = 0,
(1,
0) is a maximum turning point.
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