Prove that maximum value of a^2*b^3*c^4 subject to a+b+c=18 is 4^2*6^3*8^4. without using tat AM,GM equality thing...

You can use the method of Lagrange Multiplifiers. It is not part of IITJEE syllabus(I am not fully sure regarding this). It is easy if you use this method.

Approved

Can u pls say  what r  Lagrange Multiplifiers  in detail?

Lagrange multipliers, also called Lagrangian multipliers (e.g., Arfken 1985, p. 945), can be used to find the of a multivariate function subject to the constraint , where and are functions with continuous first partial derivatives on the open set containing the curve , and at any point on the curve (where is the gradient).

For an extremum of to exist on , the gradient of must line up with the gradient of . In the illustration above, is shown in red, in blue, and the intersection of and is indicated in light blue. The gradient is a horizontal vector (i.e., it has no -component) that shows the direction that the function increases; for it is perpendicular to the curve, which is a straight line in this case. If the two gradients are in the same direction, then one is a multiple () of the other, so

(1)

The two vectors are equal, so all of their components are as well, giving

(2)

for all , ..., , where the constant is called the Lagrange multiplier.

The extremum is then found by solving the equations in unknowns, which is done without inverting , which is why Lagrange multipliers can be so useful.

For multiple constraints , , ...,