Question

These statements are true or false?Please justify your answer.(i)The function `f(x,y)=(x³y+1,x²+y²)` is locally invertible at`(1,2)`.(ii)The function `f(x,y)=x³+y³` is integrable on `[1,2]×[1,3].`

Answer 1

See below.

Explanation:

(i)

We have

`(((partial f)/(partial x)),((partial f)/(partial y)))=((3x^2y, 2x),(x^3,2y))`

and at `(1,2)` we have

`(((partial f)/(partial x)),((partial f)/(partial y)))_((x=1),(y=2)) = ((6,2),(1,4))`

and

`det((6,2),(1,4))=22` so `(((partial f)/(partial x)),((partial f)/(partial y)))_((x=1),(y=2))` is invertible hence locally invertible.

(ii)

`f(x,y)` is a sum of integrable functions hence itself integrable.