April 16th, 2018, 11:58 PM  #1 
Member Joined: Jul 2017 From: KOLKATA Posts: 30 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: 3,261 Thanks: 894 
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 
Member Joined: Jul 2017 From: KOLKATA Posts: 30 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: 7,446 Thanks: 2499 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: 3,261 Thanks: 894 
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 
Member Joined: Jul 2017 From: KOLKATA Posts: 30 Thanks: 2 
Thanks . Got it .The R.H.L = 0 only when p>q

April 17th, 2018, 10:51 PM  #7 
Member Joined: Jul 2017 From: KOLKATA Posts: 30 Thanks: 2 
I mean R.H.L = 0 ( at x> 0+)


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