Solving 1.a using Maple

Define your f and find the partial derivatives wioth respect to x and y

>	f:=x^4+y^4-x^2; fx:=diff(f,x); fy:=diff(f,y);

Now we find the singular points:

>	solvefor[x](f,fx,fy);

find its Taylor series around the singular  point (0,0) up to the third term which is of degree 2

>	f2:=mtaylor(f,[x=0,y=0],3);

Solving 1.b using Maple

Now we define an equation g and take its partial derivatives with respect to x and y

>	g:=x^6+y^6-x*y; gx:=diff(g,x); gy:=diff(g,y);

Find the singular points

>	solvefor[x](g,gx,gy);

find its Taylor series around the singular  point (0,0) up to the third term which is of degree 2

>	g2:=mtaylor(g,[x=0,y=0],3);