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Grade 12Differential Calculus

verify rolle’s theorem for f(x) = log[(x^2 + ab )/(a+b)x ] in [a,b]
my doubt is that whether logarithmic function is continuous for all x belongs to [a,b] in this question as we don’t know what a and b are so how can we tell that its continuous ?

Profile image of shruti
10 Years agoGrade 12
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1 Answer

Profile image of Vikas TU
10 Years ago
Check condition for
f(a) = f(b)
that is true.
it comes to be zero.
Check differentiability,
 
f’(x) = 2x/(x^2 +ab) – (a+b)/(a+b)x
      = 2x/(x^2 +ab) – 1/x
  f’(a) = -b/(a+b)
 
f’(b) = (b-a)/b(b+a)
Now here,  f’(a) and f’(b) both exists at some point.
The function is differntiable and hence by property evry differntiable function will be continuos thus.
 
From Roll’s theorem then there exists c such that
f’(c) = 0
therfore, after solving c = +root(ab)  and  -root(ab)