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 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: 3,261 Thanks: 893 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: 7,341 Thanks: 2463 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: 893 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+)

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