Question

The function f: R^2 to R , defined by f(x,y)= 1-y^2+x^2, has an extremum at (0,0).

True or false with full explanation

Answer 1

ANSWER. The function $f(x,y)=1-{ y }^{ 2 }+{ x }^{ 2 }$ has no an extremum at the point (0,0).

EXPLANATION. Since $f(0,y)=1-{ y }^{ 2 }\le 1=f(0,0)$ and $f(x,0)=1+{ x }^{ 2 }\ge 1=f(0,0)$ then by the definition of the extremum of the function $f$ has no extremum at the point (0,0).