# lec4-2 - Maxima/minima with constraints • Very often we...

This preview shows pages 1–5. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Maxima/minima with constraints • Very often we want to find maxima/minima but subject to some constraint • Example: A wire is bent to a shape y = 1- x 2 . If a string is stretched from the origin to the wire, at what point along the wire is the length of the string minimized? We want to minimize d 2 = x 2 + y 2 . We can eliminate the y 2 using y 2 = (1- x 2 ) 2 , so we minimize the function f ( x ) = x 2 + (1- x 2 ) 2 = x 4- x 2 + 1 • We find df dx = 4 x 3- 2 x . The minimum occurs at 2 x 2- 1 = 0 so x = ± p 1 / 2. • We also find a local maxima at x = 0. Check with second derivatives! Patrick K. Schelling Introduction to Theoretical Methods Another approach... • Let’s do the same problem, but starting from f ( x , y ) = x 2 + y 2 , and then the differential df = 2 xdx + 2 ydy • Or we can write as, df dx = 2 x + 2 y dy dx • Then we can obtain dy dx from the equation of constraint y = 1- x 2 dy dx =- 2 x Patrick K. Schelling Introduction to Theoretical Methods Another approach continued... • Then we substitute into df dx = 0, df dx = 2 x- 4 xy = 0 • We can also solve df = 0 since dx is arbitrary, so 2 x- 4 xy = 0 • We still get 2 x- 4 x (1- x 2 ) = 0 so that x = ± q 1 2 or x = 0 as before Patrick K. Schelling Introduction to Theoretical Methods Method of Lagrange multipliers • The approaches above work, either by substituting and eliminating a variable, or by finding dy dx • However, these approaches can often lead to inconvenient algrebra • We note that we can write the constraint φ ( x , y ) = constant (sometimes we write it so that φ ( x , y ) = 0) • Then we have, for minimization of f ( x , y ) with constraint φ (...
View Full Document

{[ snackBarMessage ]}

### Page1 / 14

lec4-2 - Maxima/minima with constraints • Very often we...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online