Therefore, we can restrict our attentionto just x,y20. Now, note that f(x,y)is maximized when xy is maximized.Therefore, our problem now is,
maxxy such that x+y=8 and x,y20.
Let g(x)=x(8-x). Then, g'(x)=8-2x and g"(x)=-2. Therefore, xy is maximum at x=4,y=4. Hence, f(x,y) is maximum at xy=16, which gives maxf(x,y)=163+3×16=4144.
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Now, f(0, 0) = 0 and f(1, 1) = 4 . Thereforemaxf(x,y)≥0.
Note that, f(x,y)≥0 for x,y≥0.
And, f(-x,-y)=f(x,y) and f(-x,y)=f(x,-y)=-f(x,y).
Therefore, we can restrict our attentionto just x,y20. Now, note that f(x,y)is maximized when xy is maximized.Therefore, our problem now is,
maxxy such that x+y=8 and x,y20.
Let g(x)=x(8-x). Then, g'(x)=8-2x and g"(x)=-2. Therefore, xy is maximum at x=4,y=4. Hence, f(x,y) is maximum at xy=16, which gives maxf(x,y)=163+3×16=4144.
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