8) Constrained Maxima And Minima Of A Function

Question 1

Min max distance from origin to a surface?

Answer 1

Use the. Squared distance function subject to constraint that is the surface:

f(x,y,z) = x^2 + y^2 + z^2

E.g. Ellipse ax^2 + by^2 + cz^2 = 1.
φ(x,y,z) = ax^2 +by^2 +cz^2 -1 =0
...
..
.

Question 2

Constrained minima and maxima of a function in general

Answer 2

We may use the constraint function to express an independent variable in terms of others to find stationary points of f.

E.g. Substitute by using the constraint expressed as a function of x into f(x,y) to give g(x) and find stationary points from g’(x). And use symmetry to fully identify.
Find the value of function.

Question 3

E.g. For constraint is unit circle and function f(x,y)=xy

Answer 3

Consider first portion of the circle in the upper half plane y bigger or equal to 0. Then use constraint to express the function as a function of x. We then use symmetry for all minima and maxima.

Alternatively parameterise the constraint curve e.g.
f(x,y) =(costheta,sintheta) and function becomes sin2theta/2 hence maximal value at pi/4. And minimal at 3pi/4 and 7pi/4.

Question 4

Lagrange multipliers

Answer 4

We set the GRADIENT of the LAGRANGIAN equal to 0.

1) introduce a new variable lambda define function: L(x,y,…,lambda) = f(x,y..) - lambda( φ(x,y,…) -c)
2) set grad L(x,y,..) = 0 vector
3) consider each stationary point, value of f to see max or min

Question 5

Constraints and level sets

Answer 5

We consider level sets, requiring the highest and lowest level sets that the constraint curve touches as tangent.

If not tangents then level set would cut constraint curves at higher or lowers sets at another

Ie gradients of f and φ are linearly independent.

Question 6

Deriving lagrangian function

Answer 6

Considering a point in the level set of the constraint curve such that gradient φ at that point is not equal to 0.

As (gradφ)p is a normal vector to L_0(φ) at P. For any u tangent to L_0(φ) u_• (grad φ)_p =0z

If P is a stationary point if f_c (f defined on L_0(φ)) then the directional derivatives of f in all directions tangent to L_0(φ) must be 0.

Either we have grad f at p is 0 (stationary point of unconstrained)
OR both gradients are normal to L_0(φ) at P.

Question 7

We require that.

Answer 7

(Grad f) _(x_0,y_0) gives a vector perpendicular to contour line passing through (x_0,y_0)

Tangency condition.