Find the absolute maximum and minimum values and their locat

Find the absolute maximum and minimum values, and their locations, of the function f(x,y)=6x^2?3xy+9y+4 on the region D={(x,y)|0?x?5,0?y?14}.

Solution

f = 6x^2 - 3xy + 9y + 4

Finding partials :
fx = 12x - 3y = 0 ---> 4x = y
fy = -3x + 9 = 0 ---> x = 3

With x = 3, we get y = 12

So, only critical is (3 ,12)

Now, the endpoints of this are the lines
x = 0 , x= 5 , y = 0 and y = 14

At x = 0, we get f = 9y + 4 --> no critical
At x = 5, we get f = some linear function in y ---> no criticals
At y = 0 , we get f = 6x^2 + 4 --> fx = 12x =0 ---> x = 0
So, (0,0) is another critical
At y = 14, f = 6x^2 - 42x + 130 --> fy = 12x - 42 = 0
x = 3.5
And so another critical is (3.5 , 14)

So far we have these criticals :
(3,12)
(0,0)
(3.5,14)
(5,0)
(5,14)
(0,14)

Lets plug these and find f = 6x^2 ? 3xy + 9y + 4

(3,12) --> f = 58
(0,0), --> f = 4
(3.5,14) --> f = 56.5
(5,0) --> f = 154
(5,14) --> f = 70
(0,14) ---> f = 130

So,
abs max = 154 at (5,0)
abs min = 4 at (0,0)