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);

f := x^4+y^4-x^2

fx := 4*x^3-2*x

fy := 4*y^3

Now we find the singular points:

>    solvefor[x](f,fx,fy);

{x = 0, y = 0}

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);

f2 := -x^2

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);

g := x^6+y^6-y*x

gx := 6*x^5-y

gy := 6*y^5-x

Find the singular points

>    solvefor[x](g,gx,gy);

{x = 0, y = 0}

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);

g2 := -y*x