I have to find absolute maximum and minimum values of $f(x,y)$ = $4x^{2} + 9y^{2} -8x - 12y + 4 $
over rectangle in first quadrant bounded by lines $x=2 , y=3$ and coordinate axes
I have checked interior points for maxima and minima .But for points on the boundary i need to check now .I can also substitute values in original function of two variables and reduce it into single variable and check extremum on boundary . But im interested to use lagrange multipliers in this case . Can anyone help me with that ? Thanks in advance
Lagrange multipliers seems overkill.$$f(x, y) = 4x^2 + 9y^2 -8x - 12y + 4 = 4(x-1)^2+(3y-2)^2-4$$
Clearly it gets a global minimum of $-4$ when $x = 1, y = \frac23$ which is within the feasible region. Further, as it is convex, the maximum has to be at the boundary corners, so we need only to check $f(0, 0), f(2, 0), f(0, 3), f(2, 3)$ to find the maximum at $f(0, 3)=49$.