Question

Let f(X) be a differentiable function such that f(a)/a = f(b)/b proove that there exist a 'c' in a,b such that c*f'(X) = f(c) .

*Application of derivatives*​

Answer 1

Hi !

Solution :- let g(x) = f(x)/x

g(X) is continuious in [a,b] and differentiable in (a,b) and g(a) = f(a)/a ,

g(b) = f(b)/b

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According to given .

f(a)/a= f(b)/b

According to rolls theorem there exist a 'c' in (a,b) such that g'(c) = 0

c*f'(c) - f(c)/c² = 0

c * f'(c) = f(c )

$.$

_________________________

Hope it helps !!!

Answer 2

Step-by-step explanation:

let g(x) = f(x)/x

g(X) is continuous in [a,b] and differentiable in (a,b) and g(a) = f(a)/a ,

g(b) = f(b)/b

Given,

f(a)/a= f(b)/b

According to rolls theorem there exist a 'c' in (a,b) such that g'(c) = 0

c*f'(c) - f(c)/c² = 0

c * f'(c) = f(c )

Hope it helps1