Let f and g be differentiable functions with following properties: If h ( x )...

Jamie Medina

Jamie Medina

Answered

2022-11-26

Let f and g be differentiable functions with following properties: If h ( x ) = f ( x ) g ( x ) and h ' ( x ) = f ( x ) g ' ( x ), then what is f ( x ) ?

Answer & Explanation

Aldo Rios

Aldo Rios

Expert

2022-11-27Added 8 answers

h ( x ) = f ( x ) g ( x )
differentiate w.r.t ‘ x
h ' ( x ) = d d x f ( x ) g ( x ) = f ' ( x ) g ( x ) + f ( x ) g ' ( x ) d d x u v = u ' v + v ' u
substitute h ' ( x ) = f ( x ) g ' ( x ) in above equation
f ( x ) g ' ( x ) = f ' ( x ) g ( x ) + f ( x ) g ' ( x ) Given f ' ( x ) g ( x ) = 0
Since g ( x ) > 0 for all x is given
So f ' ( x ) = 0
It means f ( x ) is some constant
f ( x ) = C
Given f ( 0 ) = 1
Which means C = 1
Therefore f ( x ) = 1

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