Chapter 4: Q5P (page 190)

For $w = 8 x^{4} + y^{4} - 2 x y^{2}$ , find $\partial^{2} w / \partial x^{2}$ and $\partial^{2} w / \partial y^{2}$ at the points where $\partial w / \partial x = \partial w / \partial y = 0$ .

Short Answer

Expert verified

The value of provided equation at (0,0) both zero and at $(\frac{1}{4}, \frac{1}{2}) \frac{\partial^{2} w}{\partial x^{2}} = 6$ and $\frac{\partial^{2} w}{\partial y^{2}} = 2$

Step by step solution

Explanation of solution

In this question, it is provided that $w = x^{4} + y^{4} - 2 x y^{2}$ and to find $\frac{\partial^{2} w}{\partial x^{2}}$ and $\frac{\partial^{2} w}{\partial y^{2}}$ .

Partial differentiation

The procedure of calculating the partial derivative of a function is called partial differentiation. This method is used to get the partial derivative of a function with respect to one variable while keeping the other constant.

Calculation

Put $\frac{\partial^{2} w}{\partial y^{2}}$ and $c \frac{\partial w}{\partial x} = 0$

Consider,

$32 x^{3} - 2 y^{2} = 0 \to | 4 y^{3} - 4 x y = 0 \to | |$

Solve equation I and II two points, namely (0,0) and $(\frac{1}{4}, \frac{1}{2})$

Now,

$w = 8 x^{4} + y^{4} - 2 x y^{2}$

Implies that,

$\frac{\partial w}{\partial x} = 32 x^{3} - 2 x y^{2}$

Since,

$\frac{\partial^{2} w}{\partial x^{2}} = 96 x^{2}$

So, at (0,0), $\partial^{2} w / \partial x^{2} = 0$

And at role="math" localid="1659090872615" $(1 / 4, 1 / 2); \partial^{2} w / \partial x^{2} = 6$

Now,

$\frac{\partial w}{\partial y} = 4 y^{3} - 4 x y$

Implies that,

$\frac{\partial^{2} w}{\partial y^{2}} = 12 y^{2} - 4 x$

So, at (0,0) $\frac{\partial w}{\partial y} = 0$

And at the point $(1 / 4, 1 / 2);$

$\frac{\partial^{2} w}{\partial y^{2}} = 12 \times \frac{1}{4} - 4 \times \frac{1}{4}$

Therefore, at $(\frac{1}{4}, \frac{1}{2}); \frac{\partial^{2} w}{\partial y^{2}} = 2$

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For $w = 8 x^{4} + y^{4} - 2 x y^{2}$ , find $\partial^{2} w / \partial x^{2}$ and $\partial^{2} w / \partial y^{2}$ at the points where $\partial w / \partial x = \partial w / \partial y = 0$ .

Short Answer

Step by step solution

Explanation of solution

Partial differentiation

Calculation

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For w=8x4+y4-2xy2, find∂2w/∂x2and∂2w/∂y2at the points where∂w/∂x=∂w/∂y=0.

Short Answer

Step by step solution

Explanation of solution

Partial differentiation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

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For $w = 8 x^{4} + y^{4} - 2 x y^{2}$ , find $\partial^{2} w / \partial x^{2}$ and $\partial^{2} w / \partial y^{2}$ at the points where $\partial w / \partial x = \partial w / \partial y = 0$ .