Compute the second partial derivatives ∂2f ∂x2 , ∂2f ∂x ∂y , ∂2f ∂y ∂x , ∂2f ∂y2 for the following function. f(x, y) = 2xy (x2 + y2)2 , on the region where (x, y) ≠ (0, 0) Verify the following theorem in this case. If f(x, y) is of class C2 (is twice continuously differentiable), then the mixed partial derivatives are equal; that is, ∂2f ∂x ∂y = ∂2f ∂y ∂x .

Answer with step-by-step explanation:We are given that a function [tex]f(x,y)=2xy(x^2+y^2)^2[/tex]Differentiate partially w.r.t xThen, we get [tex]\frac{\delta f}{\delta x}=2y(x^2+y^2)^2+8x^2y(x^2+y^2)=(x^2+y^2)(2x^2y+2y^3+8x^2y)=2(5x^2y+y^3)(x^2+y^2)[/tex]Differentiate again w.r.t x[tex]\frac{\delta^2f}{\delta x^2}=2(10xy)(x^2+y^2)+4x(5x^2y+y^3)=20x^3y+20xy^3+20x^3y+4xy^3=40x^3y+24xy^3[/tex]Differentiate function w.r.t y[tex]\frac{\delta f}{\delta y}=2x(x^2+y^2)^2+2xy\times 2(x^2+y^2)\times 2y[/tex][tex]\frac{\delta f}{\delta y}=(x^2+y^2)(2x^3+2xy^2+8xy^2)=2(x^2+y^2)(x^3+5xy^2)[/tex]Again differentiate w.r.t y [tex]\frac{\delta^2f}{\delta x^2}=2(2y)(x^3+5xy^2)+20xy(x^2+y^2)=4x^3y+20xy^3+20x^3y+20xy^3=24x^3y+40xy^3[/tex]Differentiate partially w.r.t y [tex]\frac{\delta^2f}{\delta y\delta x}=2(2y(5x^2y+y^3)+(x^2+y^2)(5x^2+3y^2))=10x^4+36x^2y^2+10y^4[/tex][tex]\frac{\delta^2f}{\delta y\delta x}=10x^4+36x^2y^2+10y^4[/tex][tex]\frac{\delta^2f}{\delta x\delat y}=2(2x(x^3+5xy^2)+(3x^2+5y^2)(x^2+y^2))=10x^4+36x^2y^2+10y^4[/tex][tex]\frac{\delta^2f}{\delta x\delat y}=10x^4+36x^2y^2+10y^4[/tex]Hence, if f(x,y) is of class [tex]C^2[/tex] (is twice continuously differentiable), then the mixed partial derivatives are equal.i.e[tex]\frac{\delta^2f}{\delta y\delta x}=\frac{\delta^2f}{\delta x\delta y}[/tex]