Question

to find maxima and minima........
function containing 2 variables....
function containing single variable..........
lagranges method of undetermined multiples!!!!!!!!!!
explain these in simple!!!!!!
plz.

Answer 1

The strategy to find max, min is:
1. Take f' (x), then set f '(x) = 0 -> get roots r as max or min
2. Take f" (x), then plug f" ( r ) to check positive or negative,
=> determine max, min!
In case multiple variable func, apply partial derivative!

Answer 2

for single variable it's explained above
for two variables ... it's a bit tedious
you need to find ... fxx(x,y) and fxy(x,Y) and fyy(x,y)
http://www.analyzemath.com/calculus/multivariable/maxima_minima.html
It's explained well here

Answer 3

if you want to understand how this works then ... it's explained in this video
http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/part-a-functions-of-two-variables-tangent-approximation-and-optimization/session-28-optimization-problems/
or next video ....

Answer 4

Lagrange multiplies ... it is mostly used when you have minimize or maximize a function under given constraint (two equations and three variables where it seems unsolvable)
example: x^2+y^2+z^2 = 25, find max f(x,y,z) = xyz
you can find critical points ... but not determine maxima or minima, (usually you plug in those values and check if it's max or min)

this is also explained in video http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/part-c-lagrange-multipliers-and-constrained-differentials/
though optimization of two variables under a single constraint can be done with this method ... generally, but using partial derivative is better.

Answer 5

also note that
\[ f''(r+\epsilon ) \text{ and } f''(r - \epsilon )\] must have different signs (in single variable case)