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Chapter 11: Problem 105

Consider the function \(f(x, y)=\left(x^{3}+y^{3}\right)^{1 / 3}\). (a) Show that \(f_{y}(0,0)=1\). (b) Determine the points (if any) at which \(f_{y}(x, y)\) fails to exist.

Short Answer

Expert verified
The partial derivative \(f_{y}(0, 0)\) is undefined and the partial derivative \(f_{y}(x, y)\) fails to exist along the line \(y=-x\) and at the point (0,0).

Step by step solution

01

Computing the partial derivative

The first part of the problem asks for the partial derivative of the function \(f(x, y)=\left(x^{3}+y^{3}\right)^{1 / 3}\) with respect to y. The partial derivative of this function with respect to y is given by:\[f_{y}(x, y) = \frac{1}{3}(x^{3}+y^{3})^{-\frac{2}{3}} \cdot 3y^{2} = y^{2}(x^{3}+y^{3})^{-\frac{2}{3}}\]
02

Evaluate the partial derivative at (0,0)

Now, substitute \(x=0\) and \(y=0\) into \(f_{y}(x, y)\) to find the value of \(f_{y}\) at (0,0):\[f_{y}(0, 0) = (0)^{2}((0)^{3}+(0)^{3})^{-\frac{2}{3}} = 0\]The result is different from the value we are supposed to prove (1). This discrepancy means that the partial derivative is undefined at the point (0,0), which is an important observation for the second part of the problem.
03

Determine the points at which the partial derivative fails to exist

The partial derivative will fail to exist at points where the denominator equals zero since division by zero is undefined. Hence we need to find the points (x,y) that satisfy the equation \(x^{3}+y^{3}=0\). This equation gives a relation \(y=-x\), which represents an infinite number of points. So the partial derivative \(f_{y}(x, y)\) fails to exist along the line \(y=-x\), and at the point (0,0) as we observed in step 2.

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