Question

Maximum value of a 3-d plane is to be found over a circular region. Which of the following happens if we increase the radius of the circular region.

(a) Maximum value is invariant

(b) Maximum value decreases

(c) Maximum value increases and minimum value goes lesser

(d) minimum value goes higher

This question was posed to me by my college professor while I was bunking the class.

Enquiry is from Lagrange Method of Multiplier to Find Maxima or Minima in chapter Maxima and Minima of Engineering Mathematics

Answer 1

Correct choice is (c) Maximum value increases and minimum value goes lesser

For explanation I would say: Consider the level curves of the plane. These are the set of straight lines with equal slope and unequal intercepts. Now as the radius of the circular region is increased, we see that the Lagrange condition(i.e. the level curves to be tangent to the circular boundary) happens to occur further away form the origin. Thus the maximum value is pushed further and the minimum value is decreased further.