Home » Ask & Discuss » Mathematics. » Algebra « Back to Discussion

Algebra

Blazing goIITian

Joined:	17 Apr 2009
Post:	424

12 Jan 2010 10:13:17 IST

0 People liked this

361

3a-4b+45=0.minimize a^2 +b^2.ANYONE HAS OTHER METHOD THAN SOLVING FOR a or b AND THEN USING DERIVATI

None

3a-4b+45=0.minimize a^2 +b^2.ANYONE HAS OTHER METHOD THAN SOLVING FOR a or b AND THEN USING DERIVATIVE TEST? PLEASE REPLY.

Share this article on:

Comments (2)

Yagyadutt Mishra

Forum Expert

Joined: 19 Feb 2009

Posts: 1984

12 Jan 2010 12:56:01 IST

Like 1 people liked this

Yes without using derivative also you can solve it........

Method is called as Langrange multiplier....

You will study about it in detail in ur first sem............

If f(x,y) is to minimize or maximize....under the condition of g(x,y) = C............then we just write an equation like this..

f(x,y) + K (g(x,y) - c) = H(x.y)

Now by theorwm..............dH(x,y)/dx = 0

dH(x,y)/dy = 0

and dH(x,y)/dK = 0

By the help of these three equation you calculate all the critical points where f(x,y) give aximum or minimum......

For your case f(x,y) = x^2 + y^2

g(x,y) = 3x-4y

and c = -45

So equation will be

H(x,y) = x^2 + y^2 +K {3x-4y + 45}

dH(x,y)/dx = 2x + 3K = 0 or x = -3K/2 ----------(1)

dH(x,y)/dy = 2y - 4K = 0 or y = 2K -----------(2)

dH(x,y)/dK = 3x -4y +45 = 0

so 3x - 4y +45 = 0

from (1) and (2) -9K/2 -8K +45 = 0

-25K = -90

K = 90/25 = 18/5...0btain x and y from (1) and (2)

hence you got x and y..........so you have the extreme value of f(x,y)

Hari Shankar

Forum Expert

Joined: 28 Feb 2007

Posts: 2173

12 Jan 2010 15:40:51 IST

Like 3 people liked this

Two more methods

1. Interpret this geometrically. If you write x,y for a and b, the problem is minimize x^2+y^2 given that

3x-4y+45=0

In other words find the shortest distance from origin of a point (x,y) lying on the line

3x-4y+45=0

This is obviously (or u can use Triangle Inequality and prove) the perpendicular distance from origin (0,0) to this line.

This is $rac{45}{sqrt{3^2+4^2}} = 9$ and hence the minimum value of ^{is 81.}

2. You can use Cauchy Schwarz Inequality

$(3^2+4^2)(a^2+(-b)^2) ge (3a-4b)^2 = 45^2 Rightarrow a^2+b^2 ge rac{2025}{25} = 81$

Quick Reply

Reply

	Some HTML allowed.
	Keep your comments above the belt or risk having them deleted.
	Signup for a avatar to have your pictures show up by your comment
	If Members see a thread that violates the Posting Rules, bring it to the attention of the Moderator Team

Free Sign Up!

Preparing for IIT-JEE ?

Arihant Revision Package for IIT JEE - Books, Practice Tests + Rank Predictor

@ INR 1,995/-

For Quick Info

Find Posts by Topics

Physics.

Topics

9389

20086

3671

2185

7614

2826

2634

Mathematics.

Chemistry.

Biology

Institutes

Parents

Board

Fun Zone