Transcript Finding stationary points on a curve

Differentiation 2.
LO: to be able to find stationary points on a curve using differentiation.
dy
d2y
Find dx and dx 2
a.
b.
c.
d2y
4
2
dx
y  2 x  3x  1
dy
 4x  3
dx
y  2 x  5x  2 x  3
dy
 6 x 2  10 x  2
dx
2
3
2
y  x3  1
d.
y  2x
e.
3
y
x
f.
in the following:
1
2
x 1
y 3
x
dy
 3x 2
dx
1

dy
x 2
dx
dy
 3 x  2
dx
dy
 2 x 3  3x  4
dx
d2y
 12 x  10
2
dx
d2y
 6x
2
dx
d2y
1  32
 x
dx 2
2
d2y
3

6
x
dx 2
d2y
4
5

6
x

12
x
dx 2
Differentiation 2.
LO: to be able to find stationary points on a curve using differentiation.
Finding the turning points of the curve
y  x3  6 x 2  9 x  1
What is the gradient at a stationary point??
dy
0
dx
dy
 3 x 2  12 x  9
dx
3x 2  12 x  9  0
3
x2  4x2  3  0
( x  3)( x  1)  0
So, x = 1 or x = 3
Put into original equation to get y-coordinates:
Stationary points are: (1,5) and (3,1)
Differentiation 2.
LO: to be able to find stationary points on a curve using differentiation.
Sometimes we need to determine whether a stationary point is a
minimum point or a maximum point, without sketching a graph.
Maximum point
If
d2y
dx 2
is < 0, it is a maximum point.
If
d2y
dx 2
is > 0, it is a minimum point.
Minimum point
2
d y
 6 x  12
2
dx
At x = 1
At x = 3
d2y
 18
2
dx
d2y
6
2
dx
maximum
mimimum
Differentiation 2.
LO: to be able to find stationary points on a curve using differentiation.
Find the stationary points on the curve y  x  x  x  2
whether they are minimum points or maximum points.
3
and determine
d2y
 6x  2
2
dx
dy
 3x 2  2 x  1
dx
At stationary points:
2
dy
0
dx
3x 2  2 x  1  0
(3x  1)( x  1)  0
49
1
y

x
27
3
x  1
y 3
When x = 1/3,
d2y
1

6

24
2
dx
3
MINIMUM
When x = -1,
d2y
 6  1  2  4
2
dx
MAXIMUM
There is a minimum point at (1/3, 49/27)
and a maximum point at (-1,3).
Differentiation 2.
LO: to be able to find stationary points on a curve using differentiation.
Find the turning values of each function and state whether the value is a
maximum or minimum.
a) y  2 x 3  9 x 2  2
3
2
d ) y  x  2x  x  7
b) y  x 3  3x 2  10
c) y 
1 3 1 2
x  x  2x
3
2
e) y  x 
1
x
The curve y  x  ax  bx  11 has a stationary point at (3, -70).
a) Use the fact that the curve passes through (3,-70) (i.e. y = -70 when x = 3) to
find an equation involving a and b.
b) Find dy and use the fact that dy  0 when x=3 to find another equation
dx
dx
involving a and b.
c) Solve these two equations to find a and b.
d) Find d y and use this to determine whether (3, -70) is a local maximum or
dx
minimum.
3
2
2
2