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LECTURE TWO
STATIONARY POINTS
They are also referred to extreme values, critical points. A necessary condition
that a function has critical point is
fx x
f
and 0
fy y
f
…………………………………..A
If point x 0 , y 0 is a critical point satisfying equation A above and
2
( Fxx )( Fyy ) Fxy ……………………………………………..B
Then:
i. Point x 0 , y 0 is maximum point if^ ^0 ,^ fxx ^0 , fyy ^0
ii. Point x 0 , y 0 is minimum point if 0 , fxx 0 , fyy 0
iii. Point x 0 , y 0 is saddle point (undefined) if 0
iv. If 0 no information is given and therefore further investigation is
required.
EXAMPLE
From equation D we can see that x 0 & x 1. Now use the fact that
2 y x
to get the critical points.
2 x y 0 , 0 ^ ………………………………………………..E
2 x y 1 , 1 …………………………………………………..F
So, we get two critical points. All we need to do now is classify them. To
do this we will need . Here is the general formula for
2 f (^) xx x , y. fyy x , y fxy x , y
36 9
2
xy
x y
……………………………………………………..G
To classify the critical points all that we need to do is plug in the critical
points and use the fact above to classify them.
For point (^0) , 0 : 36 xy 9 36 0 0 9 9
So, for (^) 0 , 0 , is negative and so this must be a saddle point.
For point 1 , 1 : 36 xy 9 36 11 9 27 0 and
fxx 1 , 1 6 0
For ( 1 , 1 , is positive and fxx is positive and so we must have a relative
minimum.
EXAMPLE 2.
Find and classify all the critical points of , 3 3 2
2 3 2 f x y x y y y
SOLUTION
As with the first example we will first need to get all the first and second
order derivatives.
f y f y f x
f xy x f x y y
xx yy xy
x y
6 6 , 6 6 , 6
2 2
We’ll first need the critical points. The equations that we’ll need to solve
this time are,
2 2
x y y
xy x
These equations are a little trickier to solve than the first set, but once you
see what to do they really aren’t terribly bad.
First, let’s notice that we can factor out a 6 x from the first equation to get,
6 x ( y 1 ) 0
So, we can see that the first equation will be zero if x 0 or y 1. Be
careful to not just cancel the xx from both sides. If we had done that we
would have missed x 0.
b) Find the extreme values of the function
2 2
f x y , xy x y 2 x 2 y 4
f f x , y x y 1 x y
3 2
c) A rectangular box without a lid is to be made from
2
12 m of cardboard.
Find the maximum volume of such a box.
Part of assignment one 2,3,a and c
CAT 1 Q1&2 for T.I.E AND BSC. EEE
2. Construction Company has been contracted to build an auditorium rectangle
in shape and is to have a volume of 12000 square feet. It is estimated that the
annual heating and cooling cost will be sh 20 per square foot for the top and sh 40
per square foot for the front and back and sh 30 per foot for the sides. Find the
dimension of the building that will results in a minimum annual heating and
cooling cost.
CAT 1 Q1&2 for GEGIS, GIS AND BED. EEE
a) Classify extreme values of the function F x , y x y 6 x 6 y 9 xy
3 3 2 2
b) A material for constructing a box costs per square meter for top
and per square meter for sides and bottom, if the volume of the
box is 12 cubic meters. Find the dimension of the box that would
minimize the cost of material for constructing.
