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].#

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Answer 1 · 2018-02-18T21:21:15

See below.

(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.