Hey I'm doing an online course and I just can't figure out what I'm doing wrong on one of the questions.
The question is to find the discriminant of the function $$f(x,y) = 5x^2y^2 + 8x^2 + 9y^2$$
The way I solved was I found the first and second partial derivatives of the function with respect to both $x$ and $y$, and I found $f_{xy}$ as well. Then I found the critical point (in my case it ended up being $(0,0)$) and plugged them into the second derivative formulas.
Then I used the discriminant formula from my textbook: $f_{xx}f_{yy}-(f_{xy})^2$
When I solved I got the discriminant as 288, but when I input the answer it comes back as incorrect
Can someone please let me know what I am doing wrong? I have a feeling my critical point is wrong but I'm not sure what else it could be with the given function.
Best Answer
$$f(x,y) = 5x^2y^2 + 8x^2 + 9y^2$$ $$f_{xx} = \frac{d}{dx}[2 x (5 y^2 + 8)]=10y^2+16$$ $$f_{yy} = \frac{d}{dy}[2y (5 x^2 + 9)] = 10x^2+18$$ $$f_{xy} = \frac{d}{dy}[2 x (5 y^2 + 8)]=20xy\implies(f_{xy})^2=400x^2y^2$$ $$\text{discriminant } f = ((10y^2+16)(10x^2+18)) - 400x^2y^2$$ $$= (100 x^2 y^2 + 160 x^2 + 180 y^2 + 288) - 400x^2y^2$$ $$\boxed{= -300 x^2 y^2 + 160 x^2 + 180 y^2 + 288}$$ $(0, 0)$ is the correct critical point. And @ $(0, 0)$ the discriminant is $\boxed{288.}$
Make sure you got the equation down correctly, otherwise notify your Professor — the error is on their part.