Answer
a) See the explanation below.
b) See the explanation below.
Work Step by Step
a) If f has a local maximum at (a,b), then we can say that if the first-order partial derivative of $f(x,y)$ exists then $f_x(a,b)=f_y(a,b)=0$
b) If f has a critical point at (a,b), then either first-order partial derivative of $f(x,y)$ do not exist or $f_x(a,b)=f_y(a,b)=0$
