How to find absolute maximum and absolute minimum in multi-variable calculus

Question

Find the absolute maximum and absolute minimum of the function $f(x,y)=xy−7y−49x+343$ on the region on or above $y=x^2$ and on or below $y=51$.

Here is my calculation: $fx = y-49 = 0$, $y = 49$, $fy = x-7 = 0$, $x = 7$

Then I find another point which are $(0, 0)$ and $(0, 51)$. Sub it into the equation $f(x,y)$, I found that

$f(7,49) = 0$

$f(0, 0) = 343$

$f(0, 51) = -14$

so the absolute maximum is 343 and minimum is -14, but the answer is wrong, why?

Answer 1

• Well, not necessarily wrong, but you didn't check whether these points were in the domain of $f$, nor did you check the max,min of $f$ when restricted to the boundary curve which is simple.

• By simple here I mean that it can be broken up into to pieces $y = 51$ and $y = x^2$, and the corresponding restrictions are $g(y)=f(x,51)$ and $h(x)=f(x,x^2)$. Find the min,max for these functions as well. You will need to specify the domain for $g,h$ (graph the region).

• Once you get those set of min's, max's, now you are ready to compare all values and by all I mean, include the ones now that you found from the critical point analysis. I hope this helps.