Help with MaTh?

Question

Find the maximum and minimum values of the function f(xyz)=yz+xy subject to the constraints y2+z2=49 and xy=9 . Maximum value is: ?
Minimum value is :?

i woudl really preaciate your help!!

Answer 1

f(xyz) = yz+xy
xy = 9
f(xyz) = yz+9
y^2+z^2 = 49
y= sqrt(49 - z^2)
f(xyz)= zsqrt(49-z^2)
df/dz = sqrt(49-z^2)-2z^2/sqrt(49-z^2)
Setting this = 0 we get
49-z^2-2z^2=0
3z^2 = 49
z= +/- 7sqrt(3)/3
Since y^2 = 49-z^2, y= +/- sqrt( 49-49/3) = +/- 7sqrt(6)/3
Since xy=9, x = 9/y = 9/(+/- 7sqrt(6)/3)= +/- 9sqrt(6)/14

Now stick all plus values for x,y, and z into f(xyz) and then all negative values and you will have your max and min.

Answer 2

Where are you stuck? All you've done is hand us the original problem.

The constraints let you express z in terms of y and x in terms of y. Do so, and substitute those into the original function. This gives you a function in one variable (y), which you should now be able to handle ... or have a more specific question to ask.

Answer 3

from y2+z2=49 you can get z=(+-)sqrt(49-y2)
from xy=9 you can get x=9/y
in the main equation:
f=y(x+z)
f=y(9/y +- sqrt(49-y2))
f=9+-y* sqrt(49-y2)
fdach= +- (sqrt(49-y2)-y2/sqrt(49-y2))=0
so, y2=49-y2
y=+-7/sqrt(2)
x=+-9*sqrt(2)/7
z=0