1. Section 16.5 Local Extreme Values
Chapter 16
Section 16.5
Local Extreme Values

2. 𝜕𝑤 𝜕𝑥 =0 and 𝜕𝑤 𝜕𝑦 =0 and 𝜕𝑤 𝜕𝑧 =0
Critical (or Stationary) Points
A critical or stationary point is a point (i.e. values for the independent variables) that give a zero gradient. On a surface this will make the tangent plane horizontal.
Stationary points for 𝑧=𝑓 𝑥,𝑦 satisfy equations:
𝜕𝑧 𝜕𝑥 =0 and 𝜕𝑧 𝜕𝑦 =0
Stationary points for 𝑤=𝑓 𝑥,𝑦,𝑧 satisfy the equations:
𝜕𝑤 𝜕𝑥 =0 and 𝜕𝑤 𝜕𝑦 =0 and 𝜕𝑤 𝜕𝑧 =0
Extreme Points
At a stationary point a surfaced can be cupped up, cupped down or neither, this determines if the point is a local max, local min or saddle point.
Cupped Down
Cupped Up
Neither x y z x y z x y z
Saddle Point
Local Max
Local Min
Critical Values for 1 Variable
For one variable functions a local max or min could be determined by looking at the sign of the second derivative.
For a surface 𝑧=𝑓 𝑥,𝑦 there are 3 different second derivatives. How do they combine to tell you if it is cupped up or down?
Concave Down
𝑑 2 𝑦 𝑑 𝑥 2 is negative
Concave Up
𝑑 2 𝑦 𝑑 𝑥 2 is positive
Neither
𝑑 2 𝑦 𝑑 𝑥 2 is neither

3. If D is negative it is neither cupped up or down. If D is positive:
The Discriminate
For a function 𝑧=𝑓 𝑥,𝑦 the discriminate is a combination of all second derivatives that determine if the function is cupped up, down or neither. It is abbreviated with 𝐷.
If the point you are evaluating this at is a stationary point it determines if it is a local max, min or saddle point.
For a surface 𝑧=𝑓 𝑥,𝑦 :
𝐷= 𝜕 2 𝑧 𝜕 𝑥 𝜕 2 𝑧 𝜕 𝑦 2 − 𝜕 2 𝑧 𝜕𝑥𝜕𝑦 2
If D is negative it is neither cupped up or down.
If D is positive:
a. If 𝜕 2 𝑧 𝜕 𝑥 2 is positive it is cupped up.
b. If 𝜕 2 𝑧 𝜕 𝑥 2 is negative it is cupped down.
3. If D is zero can not tell might be up down or neither.
Example
Find the critical (stationary) points of the surface to the right and classify them.
𝑧= 𝑥 3 −3 𝑥 2 −9𝑥+ 𝑦 3 −6 𝑦 2
𝜕 2 𝑧 𝜕 𝑥 2 =6𝑥−6
𝜕 2 𝑧 𝜕 𝑦 2 =6𝑦−12
𝜕 2 𝑧 𝜕𝑥𝜕𝑦 =0
𝜕𝑧 𝜕𝑥 =3 𝑥 2 −6𝑥−9
𝜕𝑧 𝜕𝑦 =3 𝑦 2 −12𝑦
6𝑥−6
6𝑦−12
𝜕 2 𝑧 𝜕 𝑥 2
𝜕 2 𝑧 𝜕 𝑦 2
Point
𝜕 2 𝑧 𝜕𝑥𝜕𝑦 D
Type
Set each derivative to zero and solve.
3,0 12
-12
-144
saddle
3 𝑥 2 −6𝑥−9=0
3 𝑥 2 −2𝑥−3 =0
3 𝑥−3 𝑥+1 =0
𝑥=3 and 𝑥=−1
3 𝑦 2 −12𝑦=0
3𝑦 𝑦−4 =0
𝑦=0 and 𝑦=4
3,4 12 12
144
Local
Min
−1,0
Local
Max
-12
-12
144
Since these two equations are independent the stationary points are all the combinations of x and y.
−1,4
-12 12
-144
saddle

4. Set both equations equal to zero.
Example
Find the critical (stationary) points of the surface to the right and classify them.
𝑧= 𝑥 2 𝑦−8𝑥𝑦+ 3𝑦 2 +12𝑦
𝜕𝑧 𝜕𝑥 =2𝑥𝑦−8𝑦
𝜕𝑧 𝜕𝑦 = 𝑥 2 −8𝑥+6𝑦+12
𝜕 2 𝑧 𝜕 𝑥 2 =2𝑦
𝜕 2 𝑧 𝜕 𝑦 2 =6
𝜕 2 𝑧 𝜕𝑥𝜕𝑦 =2𝑥−8
Set both equations equal to zero.
2𝑥𝑦−8𝑦=0
𝑥 2 −8𝑥+6𝑦+12=0 2𝑦 6
2𝑥−8
This system of equations is not independent (i.e. there are x’s and y’s in both). We need to solve one and substitute into the other.
Point
𝜕 2 𝑧 𝜕 𝑥 2
𝜕 2 𝑧 𝜕 𝑦 2
𝜕 2 𝑧 𝜕𝑥𝜕𝑦 D
Type
2,0 6 -4
-16
Saddle
2𝑥𝑦−8𝑦=0
2𝑦 𝑥−4 =0
6,0 6 4
-16
Saddle
4, 2 3
Local
Min
4 3 6 8
𝑦=0
𝑥=4
𝑥 2 −8𝑥+12=0
𝑥−2 𝑥−6 =0
𝑥=2 or 𝑥=6
16−32+6𝑦+12=0
6𝑦=4
𝑦= 2 3
Points:
2,0 and 6,0
Points:
4, 2 3

5. Example
Classify the critical (stationary) points of the surface: 𝑧=16− 𝑥+2 2 − 𝑦−5 4 .
𝜕𝑧 𝜕𝑥 =−2 𝑥+2
𝜕𝑧 𝜕𝑦 =−4 𝑦−5 3
𝜕 2 𝑧 𝜕 𝑥 2 =−2
𝜕 2 𝑧 𝜕 𝑦 2 =−12 𝑦−5 2
𝜕 2 𝑧 𝜕𝑥𝜕𝑦 =0
Set to zero and solve.
𝜕 2 𝑧 𝜕 𝑦 2
𝜕 2 𝑧 𝜕𝑥𝜕𝑦
Point
𝜕 2 𝑧 𝜕 𝑥 2 D
Type
−4 𝑦−5 3 =0
𝑦−5=0
𝑦=5
−2 𝑥+2 =0
𝑥=−2 -2
−2,5 ?
The critical point is: −2,5
Because 𝐷=0 can not draw a conclusion!
How do you tell what this critical point is? Form 𝑓 −2+ℎ,5+𝑘 −𝑓 −2,5 and look at this for small values of h and k.
𝑓 −2+ℎ,5+𝑘 −𝑓 −2,5 =16− −2+ℎ+2 2 − 5+𝑘−5 4 −16=− ℎ 2 − 𝑘 4
For all small values of h and k this expression will always be negative that means that the point −2,5 is a local max since the function’s value is zero there.
If the point 𝑥 0 , 𝑦 0 is a stationary point and the discriminate 𝐷 𝑥 0 , 𝑦 0 =0, for small values of h and k consider:
If 𝑓 𝑥 0 +ℎ, 𝑦 0 +𝑘 −𝑓 𝑥 0 , 𝑦 0 is negative then the point 𝑥 0 , 𝑦 0 is a local max.
If 𝑓 𝑥 0 +ℎ, 𝑦 0 +𝑘 −𝑓 𝑥 0 , 𝑦 0 is positive then the point 𝑥 0 , 𝑦 0 is a local min.
If 𝑓 𝑥 0 +ℎ, 𝑦 0 +𝑘 −𝑓 𝑥 0 , 𝑦 0 is neither then the point 𝑥 0 , 𝑦 0 is a saddle.