If x and y are real and x+y=1, find the minimum value of 7x^2 - 4xy + 9y^2 and the values that x and y then assume</SPAN>
ok you have that and .
Now substitute the first into the second. This gives:
Now differentiate and equate to zero to find the turning point
(you can clarify that this is a minimum by differentiating again and seeing it is positive)
Now plug this into to get as the min value.
Now plug back into to get
so are the values of x and y where the minimum occurs.
Since this was listed in "Pre-Algebra and Algebra", here's how to do it without calculus:
Reduce to as bananaxxx did and then complete the square.
If x= 11/2= 5.5, x-11/2= 0 and so z= -139/2. If x is any other number, x- 11/2 is not 0 so its square is positive and z is larger than -139.2. -139.2 is the smallest possible value of z and it occurs when x= 5.5. Of course, as bananaxxx says, when x= 5.5, y= 1- 5.5= -4.5.