Chapter 4: Q4P (page 190)

If $w = x^{3} - y^{3} - 2 x y + 6$ ,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 is $\frac{\partial^{2} w}{\partial x^{2}} = 6 x and \frac{\partial^{2} w}{\partial y^{2}} = - 6 y$ .

Step by step solution

Explanation of Solution

In this question, it is provided that $w = x^{3} - y^{3} - 2 x y + 6$ and

Now, find $\frac{\partial^{2} w}{\partial x^{2}} and \frac{\partial^{2} w}{\partial y} = 0$

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

Putt $\frac{\partial w}{\partial x} = 0$ and $\frac{\partial w}{\partial y} = 0$ ,

Consider,

role="math" localid="1659089100795" $3 x^{2} - 2 y = 0 . . . (1) - 3 y^{2} - 2 x = 0 . . . (2)$

Solve equation 1 and 2, (0,0)and $(\frac{- 2}{3}, \frac{2}{3})$

Now,

$w = x^{3} - y^{3} - 2 x y + 6$

Implies that,

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

Since,

$\frac{\partial^{2} w}{\partial x^{2}} = 6 x$

Therefore, At (0,0), $\frac{\partial^{2} w}{\partial x^{2}} = 0$

And at role="math" localid="1659089309920" $(\frac{- 2}{3}, \frac{2}{3}), \frac{\partial^{2} w}{\partial x^{2}} = - 4$

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

Implies that,

$\frac{\partial^{2} w}{\partial y^{2}} = - 6 y$

So,at (0,0); $\frac{\partial^{2} w}{\partial y^{2}} = 0$

And at $(\frac{- 2}{3}, \frac{2}{3}); \frac{\partial^{2} w}{\partial y^{2}}$

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If $w = x^{3} - y^{3} - 2 x y + 6$ ,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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If w=x3-y3-2xy+6,find∂2w/∂x2 and∂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

Recommended explanations on Physics Textbooks

Modern Physics

Electromagnetism

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Circular Motion and Gravitation

Kinematics Physics

Fields in Physics

Study anywhere. Anytime. Across all devices.

If $w = x^{3} - y^{3} - 2 x y + 6$ ,find $\partial^{2} w / \partial x^{2}$ and $\partial^{2} w / \partial y^{2}$ at the points where $\partial w / \partial x = \partial w / \partial y = 0$ .