Chapter 1: Q10-7P (page 1)

Find the largest $z$ for which $2 x + 4 y = 5$ and $x^{2} + z^{2} = 2 y$ .

Short Answer

Expert verified

The maximum value of is $\sqrt{\frac{11}{4}}$ .

Step by step solution

Given Information

The given functions are $2 x + 4 y = 5$ and $x^{2} + z^{2} = 2 y$ for which maximum z are required.

Important information

Lagrange multipliers can be utilised to find the maximum and minimum of any problem.

Find the maximum z

Let $f = z^{2} = 2 y - x^{2}$ and also $ϕ (x, y) = 2 x + 4 y - 5$ .

Using the Lagrange multiplier $F (x, y) = 2 y - x^{2} + λ (2 x + 4 y - 5)$ is to be maximised.

Differentiate $F (x, y)$ with respect to y and equate to 0 and solve.

$\frac{\partial F (x, y)}{\partial y} = 2 + 4 λ$

$2 + 4 λ = 0 λ = - \frac{1}{2}$

Differentiate $F (x, y)$ with respect to x and equate to 0 and solve using $λ = - \frac{1}{2}$ .

$\frac{\partial F (x, y)}{\partial x} = - 2 x + 2 λ$

$- 2 x + 2 λ = 0 - 2 x + 2 (- \frac{1}{2}) = 0 x = - \frac{1}{2}$

Substitute $- \frac{1}{2}$ for x into $2 x + 4 y = 5$ and solve to find the value of y.

$2 (- \frac{1}{2}) + 4 y = 5 - 1 + 4 y = 5 y = \frac{3}{2}$

Substitute $- \frac{1}{2}$ for x and $\frac{3}{2}$ for y into $z^{2} = 2 y - x^{2}$ and solve to find the value of z.

$z^{2} = 2 (\frac{3}{2}) - {(- \frac{1}{2})}^{2} z^{2} = 3 + \frac{1}{4} z = \pm \sqrt{\frac{11}{4}}$

It can be observed that the maximum value of z is $\sqrt{\frac{11}{4}}$ .

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Find the largest $z$ for which $2 x + 4 y = 5$ and $x^{2} + z^{2} = 2 y$ .

Short Answer

Step by step solution

Given Information

Important information

Find the maximum z

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Find the largest zfor which2x+4y=5 and x2+z2=2y.

Short Answer

Step by step solution

Given Information

Important information

Find the maximum z

One App. One Place for Learning.

Most popular questions from this chapter

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Electrostatics

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Find the largest $z$ for which $2 x + 4 y = 5$ and $x^{2} + z^{2} = 2 y$ .