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Let f(x) and g(x) be real valued functio...

Let f(x) and g(x) be real valued functions such that f(x)g(x)=1, `AA x in R."If "f''(x) and g''(x)" exists"AA x in R and f'(x) and g'(x)`
are never zero, then prove that `(f''(x))/(f'(x))-(g''(x))/(g'(x))=(2f'(x))/(f(x))`

Text Solution

Verified by Experts

`f(x)xxg(x)=1`
Differentiating w.r.t. x on both sides, we get
`f'g+fg'=0" …(1)"`
`rArr" "(g)/(g')=-(f)/(f')" …(2)"`
Differentiating (1), w.r.t x we get
f''g+2f'g'+fg''=0
`rArr" "(f''g)/(f'g')+2+(fg'')/(f'g')=0`
`rArr" "(f'')/(f')(-(f)/(f'))+2+((f)/(f'))(g'')/(g')=0" (Using (2))"`
`rArr" "(f''(x))/(f'(x))-(g''(x))/(g'(x))=(2f'(x))/(f(x))`
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