April 16th, 2018, 11:58 PM  #1 
Newbie Joined: Jul 2017 From: KOLKATA Posts: 29 Thanks: 2  Calculus  Lagrange
if f(x) = (x^p)/(sinx)^q when 0 < x < 90 = 0 when x = 0 Then prove that L agrange’s mean value theorem is applicable to f(x) in closed interval [0, x] only when p >q . Could u please put some hint 
April 17th, 2018, 03:05 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,195 Thanks: 872 
The mean value theorem, applied to interval [0, x] requires that f be differentiable on the open interval (0, x) and continuous on the closed interval [0, x]. Can you show that? (Does the problem really say "0< x< 90"? That "90" looks like 90 degrees and it is very unusual to use degrees rather than radians in Calculus) 
April 17th, 2018, 04:21 AM  #3 
Newbie Joined: Jul 2017 From: KOLKATA Posts: 29 Thanks: 2 
U are correct this is in radians ( pi/2) . However , could u please help me proving that Lagrange's MVT applies to the above only when p> q

April 17th, 2018, 05:16 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,313 Thanks: 2447 Math Focus: Mainly analysis and algebra 
You should focus on that condition of continuity. What can you tell about the continuity of $f(X)$ at $x=0$?

April 17th, 2018, 06:18 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,195 Thanks: 872 
What is $\displaystyle \lim_{x\to 0}\frac{x^p}{sin^q(x)}$ for different values of p and q? Notice that: if p= q you can write the function as $\displaystyle \left(\frac{x}{sin(x)}\right)^p$. If p> q, taking r= p q, you can write the function as $\displaystyle \left(\frac{x}{sin(x)}\right)^q x^r$. If p< q, taking r= q r, you can write the function as $\displaystyle \left(\frac{x}{sin(x)}\right)^q\left(\frac{1}{sin( x)}\right)^r$. Last edited by Country Boy; April 17th, 2018 at 06:42 AM. 
April 17th, 2018, 09:24 PM  #6 
Newbie Joined: Jul 2017 From: KOLKATA Posts: 29 Thanks: 2 
Thanks . Got it .The R.H.L = 0 only when p>q

April 17th, 2018, 10:51 PM  #7 
Newbie Joined: Jul 2017 From: KOLKATA Posts: 29 Thanks: 2 
I mean R.H.L = 0 ( at x> 0+)


Tags 
calculus, lagrange 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Lagrange Multipliers  yo79  Math Events  2  March 30th, 2013 01:00 AM 
Prove this (by using Lagrange)  OriaG  Calculus  1  January 28th, 2013 12:17 PM 
lagrange  laurens535  Calculus  1  October 29th, 2012 02:44 PM 
Lagrange Multipliers  ziggy41  Calculus  5  October 22nd, 2010 09:31 PM 
The Lagrange basis  rebecca  Linear Algebra  0  December 15th, 2009 11:46 AM 