Find the exact extreme values of the function.. z=f(x,y)=(x-15)^2+(y-15)^2 subje

Question

Find the exact extreme values of the function..

z=f(x,y)=(x-15)^2+(y-15)^2

subject to the constraints

x? (greater than or equal to) 0, y (greater than or equal to) 0

15x+15y (less than or equal to) 225

fmin = ? at (x,y) = (__,__)

fmax=550 at (x,y)= (0,0)

I am having trouble finding the fmin as when I use the boundary points of (0,15), (15,0) they give back the wrong answer. When I did partials of the f(x,y) function i found x=15 and y=15 for critical points, and when placed within the bounds, give me the same 2 points of (0,15), (15,0). I tried plugging in the linear function restraint into f(x,y) and finding the derivative with respect to x but had no luck there either. PLEASE HELP !!

Explanation / Answer

U dont need to do any derivatives

As the required function which has to be maximise or minimize is equation of circle with centre (15,15)

And the line equation 15x + 15 y =225

U need to find perpendicular from centre of circle to that line

Equation of perpendicular line is y=x

and the required area comes under x>=0 & y>=0 & 15x+15y<=225

So the perpendicular line intersects this intersected region at two place (0,0) & (7.5,7.5)

So fmax at (0,0) = (0-15)^2 + (0-15)^2 =450 (ans)

And fmin at (7.5,7.5) = (0-7.5)^2 + (0-7.5)^2 = 112.50 (ans)

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