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

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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 $\partial w / \partial x = \partial w / \partial y = 0$ .