Answer
a. $\frac{\partial z}{\partial x}=f_{xx}(x,y)=18x+8y$
$\frac{\partial z}{\partial x}=f_{xy}(x,y)=8x$
b. $\frac{\partial z}{\partial y}(-1,4)=-4$
$\frac{\partial z}{\partial y}(-1,4)=-8$
c. $f_{xy}(2,-1)=16$
Work Step by Step
We are given $z=f(x,y)=3x^{3}+4x^{2}y-2y^{2}$
$f_{x}(x,y) =9x^{2}+8xy$
$f_{y}(x,y)=4x^{2}-4y$
a. $\frac{\partial z}{\partial x}=f_{xx}(x,y)=18x+8y$
$\frac{\partial z}{\partial x}=f_{xy}(x,y)=8x$
b. $\frac{\partial z}{\partial y}=f_{yy}(x,y)=-4$
$\frac{\partial z}{\partial y}=f_{yx}(x,y)=8x$
$\frac{\partial z}{\partial y}(-1,4)=8(-1)=-8$
$\frac{\partial z}{\partial y}(-1,4)=-4$
c. $f_{xy}(2,-1)=8x=8(2)=16$
