Minimize N(x,y)=3xsquared+ysquared+2xy-54+60, condition function y=4x

To minimize the function N(x,y) = 3x^2 + y^2 + 2xy - 54x + 60 with the constraint y = 4x, we can substitute y = 4x into the function N(x,y).

Substitute y = 4x into N(x,y) gives:
N(x) = 3x^2 + (4x)^2 + 2x(4x) - 54x + 60
N(x) = 3x^2 + 16x^2 + 8x^2 - 54x + 60
N(x) = 27x^2 - 54x + 60

Now to minimize N(x), we can find the critical points by taking the derivative with respect to x and setting it to zero:
\frac{dN(x)}{dx} = 54x - 54
Setting the derivative equal to zero:
54x - 54 = 0
x = 1

Now, we need to check if this is a minimum by confirming the second derivative is positive:
\frac{d^2N(x)}{dx^2} = 54 > 0

Therefore, x = 1 minimizes the function N(x) under the given constraint.

Substitute x = 1 into y = 4x gives:
y = 4(1) = 4

So, the minimum value of N(x,y) under the given constraint occurs at (1, 4), and the minimum value is:
N(1,4) = 27(1)^2 - 54(1) + 60 = 33

\boxed{N(1,4) = 33}