Question

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=y^{3}-4 x y^{2}-1 $$

Short answer

The four second partial derivatives for the given function are \(\partial^2 z/\partial x^2 = 0\), \(\partial^2 z/\partial y^2 = 6y - 8x\), \(\partial^2 z/\partial x\partial y = -8y\), and \(\partial^2 z/\partial y\partial x = -8y\). The mixed second partial derivatives are equal.

Step-by-step solution

First Partial Derivatives

Take the first partial derivatives with respect to both \(x\) and \(y\): \[\partial z/\partial x = 0 - 4y^2 - 0 = -4y^2\] and \[\partial z/\partial y = 3y^2 - 4x(2y) = 3y^2 - 8xy\].

Second Partial Derivatives

Take the second partial derivative of both the derivatives obtained from the previous step with respect to \(x\) and \(y\) again.We get: \[ \partial^2 z/\partial x^2 = \partial (-4y^2)/\partial x = 0\], \[\partial^2 z/\partial y^2 = \partial (3y^2 - 8xy)/\partial y = 6y - 8x\], \[\partial^2 z/\partial x\partial y = \partial (3y^2 - 8xy)/\partial x = -8y\] and \[ \partial^2 z/\partial y\partial x = \partial (-4y^2)/\partial y = -8y\].

Verifying Mixed Partial Derivatives

The task also requires to check whether the mixed second partial derivatives are equal. As evident from Step 2, both the mixed partial derivatives \(\partial^2 z/\partial x\partial y\) and \(\partial^2 z/\partial y\partial x\) are equal, i.e., \(-8y\). Hence, the second mixed partials are equal.