Q10-14P Do Example 3b using Lagrange mul... [FREE SOLUTION]

Chapter 1: Q10-14P (page 1)

Do Example 3b using Lagrange multipliers.

Short Answer

Expert verified

Hence, the points on the given surface nearest to the origin are $(0, 0, 1) and (0, 0, - 1)$ .

Step by step solution

Lagrange's Multipliers

Lagrange's multipliers is a method of optimization to find the local maxima and minima of the function $f (x, y, z)$ with constraint $蠁$ , which is given by:

$F (x, y, z) = f + 位蠁$

Find derivatives

The given functions are:

$f (x, y, z) = x^{2} + y^{2} + z^{2} 蠁 (x, y, z) = z^{2} - x^{2} = 1$

Now, using Lagrange's multipliers method, we get:

$F (x, y, z) = f + 位蠁 = x^{2} + y^{2} + z^{2} + 位 (z^{2} - x^{2})$

To find the minimum or maximum angle, we have:

$\frac{鈭� F}{鈭� x} = 2 x - 2 x 位 = 0 \frac{鈭� F}{鈭� y} = 2 y = 0 \frac{鈭� F}{鈭� z} = 2 z + 2 z 位 = 0$

Find the values of variables

From $\frac{鈭� F}{鈭� x} = 0$ , we get:

$x = 0 or 位 = 1 鈬� z = 卤 1 or z = 0, x = 卤 i$

From $\frac{鈭� F}{鈭� y} = 0$ , we get:

$y = 0$

From $\frac{鈭� F}{鈭� z} = 0$ , we get:

$z = 0 or 位 = - 1 鈬� x = 卤 i or x = 0, z = 卤 1$

Taking the real values of points, the point of interest obtained is role="math" localid="1664881443001" $(0, 0, 卤 1)$ .

Hence, the points on the given surface nearest to the origin are $(0, 0, 1) and (0, 0, - 1)$ .

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91影视

Do Example 3b using Lagrange multipliers.

Short Answer

Step by step solution

Lagrange's Multipliers

Find derivatives

Find the values of variables

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