Answered: Solve the optimization problem. HINT…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Solve the optimization problem. HINT [See Example 2.]

Minimize

F = x² + y²

with

x + 2y = 10.

F =

Expert Solution

Step 1

Given minimize $F = x^{2} + y^{2}$ with $x + 2 y = 10$ .

We have to solve the optimization problem.

From equation $x + 2 y = 10$ , we get

$x = 10 - 2 y$

Substitute $x = 10 - 2 y$ in $F = x^{2} + y^{2}$

$\begin{matrix} F = {(10 - 2 y)}^{2} + y^{2} \\ = 100 + 4 y^{2} - 40 y + y^{2} \\ = 100 + 5 y^{2} - 40 y \end{matrix}$

Hence, $F = 100 + 5 y^{2} - 40 y$

Now, differentiate $F = 100 + 5 y^{2} - 40 y$ with respect to $y$ , we get

$\begin{matrix} \frac{d F}{d y} = \frac{d}{d y} (100 + 5 y^{2} - 40 y) \\ = 0 + 10 y - 40 \\ = 10 y - 40 \end{matrix}$

Hence, $\frac{d F}{d y} = 10 y - 40$ .

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One of your classmates makes the claim that the solution of an optimization problem modeled by a quadratic function occurs at the vertex of the quadratic. What is wrong with the statement? Give an example showing why this statement is wrong.
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Solve the optimization problem. HINT [See Example 2.] Minimize F = x2 + y2 with x + 2y = 10. F =