Question

How did Gandhiji deduce values? (Rewrite the
Sentence into Assertive)​

Answer 1

Lagrange Multipliers

In many applied problems, a function of three variables, f(x, y, z), must be optimized

subject to a constraint of the form g(x, y, z) = c.

Theorem: (Lagrange’s Theorem)

Suppose that f and g are functions with continuous first-order partial derivatives and f has

an extremum at (x0, y0, z0) on the smooth curve g(x, y, z) = c. If ∇g(x0, y0, z0) 6= ~0, then

there is a number λ such that

∇f(x0, y0, z0) = λg(x0, y0, z0).

The number λ is called a Lagrange multiplier.

Method of Lagrange Multipliers:

To find the extreme values of f(x, y, z) subject to the constraint g(x, y, z) = c,

1. Find all values of x, y, z, λ such that

∇f(x, y, z) = λ∇g(x, y, z),

g(x, y, z) = c.

2. Evaluate f at each point (x, y, z) found in step 1. The largest of these values is the

maximum value of f and the smallest is the minimum value of f.

Example: Find the maximum and minimum values of f(x, y) = 6x+8y on the circle x

2+y

2 =

25.

Let g(x, y) = x

2 + y

2

. The gradient vectors of f and g are

∇f(x, y) = h6, 8i and ∇g(x, y) = h2x, 2yi.

By Lagrange’s Theorem, there is a number λ such that

h6, 8i = λh2x, 2yi = h2λx, 2λyi.

Therefore, we consider the system

6 = 2λx,

8 = 2λy,

x

2 + y

2 = 25.