Find the four second partial derivatives. Observe that the second mixed partials are equal. $z = x ^ { 2 } + 8 x y + 6 y ^ { 2 }$
A) $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } = 2 , \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } = 6 , \frac { \partial ^ { 2 } z } { \partial x \partial y } = \frac { \partial ^ { 2 } z } { \partial y \partial x } = 0$
B) $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } = 0 , \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } = 6 , \frac { \partial ^ { 2 } z } { \partial x \partial y } = \frac { \partial ^ { 2 } z } { \partial y \partial x } = 8$
C) $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } = 2 , \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } = 12 , \frac { \partial ^ { 2 } z } { \partial x \partial y } = \frac { \partial ^ { 2 } z } { \partial y \partial x } = 8$
D) $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } = 0 , \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } = 0 , \frac { \partial ^ { 2 } z } { \partial x \partial y } = \frac { \partial ^ { 2 } z } { \partial y \partial x } = 8$
E) $\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } = 0 , \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } = 0 , \frac { \partial ^ { 2 } z } { \partial x \partial y } = \frac { \partial ^ { 2 } z } { \partial y \partial x } = 0$

Question

Quizplus · Accepted Answer

The second partial derivatives of $z = x^2 + 8xy + 6y^2$ are: $\frac{\partial^2 z}{\partial x^2} = 2$ (from $x^2$), $\frac{\partial^2 z}{\partial y^2} = 12$ (from $6y^2$), and $\frac{\partial^2 z}{\partial x \partial y} = \frac{\partial^2 z}{\partial y \partial x} = 8$ (from $8xy$), showing that the mixed partials are indeed equal.

Find the Four Second Partial Derivatives z=x2+8xy+6y2z = x ^ { 2 } + 8 x y + 6 y ^ { 2 }z=x2+8xy+6y2

Find the Four Second Partial Derivatives $z = x ^ { 2 } + 8 x y + 6 y ^ { 2 }$