Test for maximum, minimum and points of inflexion 1. Find stationary point a : f ' (a) = 0 2. Study the sign of f' to the right and left of a + + + maximum f ' (a)=0 horizontal point of inflexion f ' (a)=0 f ' (a)=0 + minimum f ' (a)=0
May 25, 2015
Test for maximum, minimum and points of inflexion
1. Find stationary point a : f ' (a) = 0
2. Study the sign of f ' to the right and left of a
+
+
+maximum
f ' (a)=0
horizontalpoint of inflexion
f ' (a)=0f ' (a)=0
+ minimum
f ' (a)=0
Find the coordinates of the stationary points on the curve y= x4 4 x3 and determine their nature.
Example 5:
Find the stationary points for the curve and determine their nature.Sketch the function.
Example 6:
Find turning points for . Sketch the curve.
Example 7:
For the curve y = x3 + x2 find the stationary points and sketch the curve.
Example 8:
y = x3 + x2
By the end of the lesson you will be able to:
• Use the second derivative to test the nature of a stationary point and/or point of inflexion.
•
•
A
B
f is concave upwards
gradient is increasing
f ' is increasing
f '' > 0
If a function is increasing then its derivative is positive
Concavity
f is concave downwards
gradient is decreasing
f ' is decreasing
f '' < 0
If a function is decreasing then its derivative is negative
•
• A
B
f is concave upwards f '' > 0
f is concave downwards f '' < 0
The second derivative and turning points
•
•
B
A
at A: f ' (a)= 0f is concave up. f ''(a) > 0
f '(a) = 0
A is a minimum point
f ' (b)= 0f is concave down.
f '(b) = 0f ''(b) < 0
B is a maximum point
at B:
P is a minimum point
at P :
f '(x) = 0
f ''(x) > 0
P is a maximum point
at P :
f '(x) = 0
f ''(x)< 0
P•
A point of inflexion is when the curve changes from concave down to concave up or vice versa.
P is a point of inflexion
• P
at P:
f ''(x) = 0 f '' changes sign
If f ' is also zero then P is a horizontal point of inflexion.
Find the points of inflection of f (x) = x4 4 x3 + 5
Find the stationary points for the curve y= 4 + 3x x3 and determine their nature. Draw a sketch of the function.
y= 4 + 3x x3
Find the point of inflexion on the curve y = 2 x 3 + 3 x 2 + 6 x 7 and sketch its graph
y = 2 x 3 + 3 x 2 + 6 x 7
For the curve y = x3 + x2 find the stationary points and point of inflexion.
y = x3 + x2
Find turning points for y =x3 3x2 9x +2. Sketch the curve.
y =x3 3x2 9x +2
Test for maximum, minimum and points of inflexionMethod 1: First derivative Sign test
1. Find a : f ' (a) = 0
2. Study the sign of f ' to the right and left of a
++ maximum minimum
++
f ' (a)=0
horizontalpoint of inflexion
Method 2 : Second derivative test
1. Find a : f ' (a) = 0
2. Study the sign of f ''(a)
f '' (a) <0
f '' (a) = 0 andf '' changes sign at a
f ' (a)=0f ' (a)=0
f ' (a)=0f '' (a) >0 minimum
maximum
point of inflexion
http://www.mindomo.com/view.htm?m=7af63ca6ad1b462fa05b7762859f45d3
Derivative puzzle 1
Big derivative puzzle