Find the absolute minimum and absolute maximum of f(x, y)=8−5x+9y on the closed triangular region with vertices (0,,0) and (9,12). list the minimum/maximum values as well as the point(s) at which they occur. if a min or max occurs at multiple points separate the points with commas.

maximum value is 8 and Point of maxima is (0, 0)

Minimum value is -145 and Point of maxima is (9, 12)

Step-by-step explanation:

Given;

Function:

f( x , y ) = 8 - 5x + 9y

Point of vertices ( x , y )

(0,0) , (9,0) and (9,12)

at (0,0)

f( 0 , 0 ) = 8 - 5(0) + 9(0) = 8

at (9,0)

f( 9 , 0 ) = 8 - 5(9) + 9(0)

f(9,0) = 8 - 45 - 0 = - 37

at (9,12)

f( 9 , 12 ) = 8 - 5(9) + 9(12)

f(9,0) = 8 - 45 - 108 = - 145

Hence,

maximum value is 8 and Point of maxima is (0, 0)

Minimum value is -145 and Point of maxima is (9, 12)

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hello from mrbilldoesmath!

answer:

(7 x + 6 y) (49 x^2 - 42 x y + 36 y^2)

discussion:

the trick is to recognize that 343 = 7^3 and 216 = 6^3 and, in general,

a^3 + b^3 =   (a + b) (a^2 - a b + b^2).

in our case

343x^3+216y^3   =   (7x)^3 + (6y)^3

so take a = 7x and b = 6y in the above equation to get the answer.

you,

mrb