    Question

# Let f:R+→R+ be a differentiable function satisfying f(xy)=f(x)y+f(y)x for all x,yϵR+. Also f(1)=0,f′(1)=1. If M is the greatest value of f(x) then [m+e] is ___ (where [.] represents Greatest Integer Function).

A

3

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B

2

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C

0

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D

1

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Solution

## The correct option is A 3 f(xy)=f(x)y+f(y)x f′(xy)⋅(y+xy′)=f′(x)y−f(x)y′y2+f′(y)y′x−f(y)x2 Let x=1;f′(y)⋅(y+y′)=f′(1)y−f(1)y′y2+f′(y)y′1−f(y)1 Substituting the value we can write dydx+yx=1x2. Solving the linear differential equation we get f(x)=logxx as f(1)=0. Maximum value of f(x)=1e ∴[1e+e]=3  Suggest Corrections  0      Similar questions  Related Videos   Methods of Solving First Order, First Degree Differential Equations
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