# Absolute maxima of multivariable function

hey guys this is a question from homework and I stuck on it.

Find the absolute maximum and absolute minimum of the function $$f(x,y) = xy-7y-49x+343$$ on or above $$y = x^2$$ and on or below $$y = 53$$.

The following is what I tried:

critical points inside the region:
$$f_x = y - 49$$ $$f_y = x - 7$$
so $$(7,49)$$ is a critical point with function value $$0$$.
critical points on $$y = 53$$:
$$f(x,53) = 4x-28$$
min value on the line: $$f(-53^(0.5),53) = -57.12043956$$
max value on the line: $$f(53^(0.5),53) = 1.120439557$$
critical points on $$y=x^2$$:
$$f(x,x^2) = x^3-7x^2-49x+343\\ f'(x,x^2) = 2x^2-14x-49$$ from $$f'(x,x^2) = 0$$ we can get $$x = 9.562177826$$ or $$x = -2.562177826$$ with function values respectively $$109.7266433$$ or $$405.7733568$$

so the absolute maximum should be $$405.7733568$$ but this is wrong. So what is the correct way to solve it?

It is the correct way to solve it. You just have a mistake in the last derivative. The coefficient of $$x^2$$ should be $$3$$ not $$2$$