33Or we have only one choice evaluate the indefinite by parts and take limits at last.
Given, (LL=f(y),UL=f(x))∫et dt = (LL=y,UL=x) ∫1/t dt, for all x,y belonging to (1/e2,∞) where f is continuous and differentiable function and f(1/e)=0.
If g(x)={ ex, x≥k
{ ex2, 0<x<k
Find k such that f(g(x)) is continuous for all x>0.
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20 Answers
62hmm.. ok
I think there is a mistake in typing or something's still missing cos of partial typing!
11I like this one concept..
Good
Thank you abhishek and nishant for the question and solution...
62exactly..
no a,b,c,d....
just take their integrals..
and after taht..value 0 and 1!
take teh difference..
62yeah i knew this is the doubt u will have..
give me 1 sec.. i will post another image :)
33
62x<0 it is 1/3-x/2
x>1 it is x/2-1/3
in between it is x3/3+1/3-x/2
the trick is that in between (0,1) case c above.. in my post..
u should take integral as sum of integrals from (0,x) and from (x,1)
May be i made some error in the calculation.. but the basic idea is correct... if u do get some different answer tell me.. i will try to correct it :)
62Q2
we have to take 3 cases:
a) x<0
b)x>1
c) x ε [0,1]
I hope this is a hint! But i will try to post the graph right away!
I hope u guys try to get the middle paths
33pata nahi i'm also getting nothing it is in CPP of FIITJEE sir, which i'm solving
62if is a continuous fn for all x except 0 and (LL=0,UL=a)∫f(x)dx=b
If g(x)=(LL=x,UL=a)∫f(t)/t dt, then express (LL=0,UL=a)∫g(x)dx in terms of b
yaar this question does not seem to make too much sense to me!!
Why? bcos we know nothing about f(x)!
Case I
say a=1
f(x)={x} {fractional part of x}
Case II
a=1
f(x) = 1-{x}
B is same for both these cases!
33Q.2 Let f(x)= (LL=0,UL=1)∫|t-x|*t dt for all real x. Sketch the graph of f(x) and what is the min value of f(x).
33f is a continuous fn for all x except 0 and (LL=0,UL=a)∫f(x)dx=b
If g(x)=(LL=x,UL=a)∫f(t)/t dt, then express (LL=0,UL=a)∫g(x)dx in terms of b
62hmm.. i understand what "inspires" u to do differentiate.....
But i strongly think this is not needed in this question...
The first look does not make me feel that there will be different results!
But i am not sure.. i ll check and see :)
33what if i differenciate both side of (LL=f(y),UL=f(x))∫et dt = (LL=y,UL=x) ∫1/t dt wrt x then what we will get ...
62Hey priyam, have u figured out what f(x) is?
This is simple to do using y=1/e
and the limit of integral from 1/e to x!
so u will get f(x)
The 2nd part is finding k...
If this is the actual blocking point then i will tell u that part as well :)