Q if p=(x2-x1)2+(2√x2-x1-2)2 , x1,x2 ε R and x2 > 0 , then find minimum value of p ??
11
p = (x2 - x1)2 + (2√x2 - x1 - 2)2
p' = -2(x2 - x1) - 2(2√x2 - x1 - 2) = 0 [ p' is ∂p/∂x1 ]
→ (x1 - x2) = (2√x2 - x1 - 2)
→ x1 = √x2 + x2/2 - 1
p = (x2/2 - √x2 + 1)2 + (√x2 - x2/2 - 1)2
= 2(x2/2 - √x2 + 1)2
Let √x2 = x
p = (x2 - 2x + 2)2/2
p'=0
→ (x2 - 2x + 2)(x - 1) = 0
→ x = 1
So, pmin = 1/2
21OH O k k..... read the latter partr of ur point later!!
But hey how do u know exactly these sub will work?
21followin is jus an eg.
If P = (2x1 - x + 2)2 + ( x2 - 3x1 + 5)2
then wat wud hav bn the curves?
1\texttt{The terms in the second bracket are } 2.\sqrt{x_{2}} \texttt{ and } x_1+2 \texttt{ respectively thats why !}
21yah but y do u consider jus da 2nd bracket?
mayb my que in POST # 25 will make my dbt more clear
21@Nishant Sir, Sir I m not gettin the ans wid partial differentiation approach [2] can u help me proceed
62@tapan..
if you keep x1 constant..
then it will be a quadratic in x2
you can find the x2 for minimum value in terms of x1
substitute that in original form
now minimise this for the single varaible x1
21Ya sir, i know how to partial differentiate,
"@Nishant Sir, Sir I m not gettin the ans wid partial differentiation approach can u help me proceed"
wait 2 min, I'll post my workin
21Does the min value of x in abov graf give the necessary value of P?
then how cum is it not 0.5
21Oh k! yah i got how to find the min. dis b/w these 2 curves..........
but had i doubt that WHY/HOW does making such assumptions of these 2 specific curves help u get da min val of P?
62yes tapan.. you are right.. my method gives a very very very very very dirty answer for this one..
rajat's method is very good for this one..
basically that is what should be done..
I dont see another way out!
62oops rkrish solved it :D
wow.. and i am posting that this is getting messy ;) :D
11.....BUT bhaiyya ....I worked out acc. to wat you said(partial derivative wrt x1) and got the ans.
Pls tell me if I have made any careless or silly mistake in post#33.
21OH K!!!!
thnx rkrish n SIR ( for a nice methd)
SIR : Can u pl. explain me the rational b/h Rajat's solution! (i.e. can u pl. tell me how can v apply Rajat's methd if the questn was like in POST #25
11@tapan...
I think I understood rajat's approach...
Acc. to the ques,
p=(x2 - x1)2 + (2√x2 - (x1+2))2 = d2 (let)
If you take y2 = 2√x2 : C2 (let)
& y1 = (x1+2) : C1 (let)
you get d in the form of :-
d = √(x2-x1)2 + (y2-y1)2
Now, d is minimum distance b/w curves C1 & C2
and pmin = d2
Now see if you can convert post#25 in the form of the "Dist. Formula" by some substitution.....Otherwise......no other way out(follow wat Nishant bhaiyya's approach)!!!!
21THNX rkrish........... thnx a ton buddy!!!
NOW I AM RICH WITH 2 methods of solving this type of sums........
thnx all!!!!!!!!! [1]
1ans to a hai but method bhi batao zara mane thode lambe method se kiya tha
62Vishal which si the question?
(x2-x1)^2+(2\sqrt{x2}-x1-2)^2 \\ or\\ (x2-x1)^2+(2\sqrt{x2-x1}-2)^2 \\
21maine ANS nahi dejha hai, but for sure then its gotta be less than 1....
WAS IT an MCQ?
Wat wer da options?
