Question

Evaluate the second partial derivatives $f_{x{x}^{\prime }}{f}_{x{y}^{\prime }}{f}_{y{y}^{\prime }}$ and $f_{yx}$ at the point. $$f(x,y)=x^4-3x^2{y}^2+y^2$$

Short answer

The second partial derivatives of the given function are:$f_{xx}=12x^2-6y^2$, $f_{xy}=-12xy$, $f_{yx}=-12xy$, $f_{yy}=-6x^2+2$.

Step-by-step solution

Calculate the first derivative

First, calculate the first derivative with respect to x and y:\n $f_x=4x^3-6x{y}^2$ and $f_y=-6x^{2}y+2y$.

Calculate the second derivatives

Now, proceed to calculate the second derivatives based on the first derivatives:\n $f_{xx}=\frac{\partial f_x}{\partial x}=12x^2-6y^2$, $f_{xy}=\frac{\partial f_x}{\partial y}=-12xy$, $f_{yx}=\frac{\partial f_y}{\partial x}=-12xy$ and $f_{yy}=\frac{\partial f_y}{\partial y}=-6x^2+2$.

Checking if the function is continuous

These second order partial derivatives exist for all points since the initial function f(x,y) is polynomial. Therefore it's everywhere continuous and differentiable.