Question

Find the points on the plane x + y + z = 9 which are closest to origin.

(a) (3,3,3)

(b) (2,1,3)

(c) (2,2,2)

(d) (3,4,1)

The question was asked in a national level competition.

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

Answer 1

Correct choice is (a) (3,3,3)

Easiest explanation: The objective function is f(x,y,z) = x^2 + y^2 + z^2

compute gradient ∇f = 2x i + 2y j + 2z k

Now compute gradient of the function x + y + z = 9

which is

= i + j + k

Using Lagrange condition we have

∇f = λ . ∇g

2x i + 2y j + 2z k = λ * (i + j + k)

⇒ x = y =z

Put this back into constraint function we get

3x = 9 ⇒ (x,y,z) = (3,3,3).