Expert: Ahmed Salami - 5/26/2010

Question
f(x)=-1+Kx+K neither touches nor cuts the curve f(x)=ln x, then the minimum value of K is what ?

Answer
Hi Debojyoti,
Consider the functions;
f(x) = Kx + (K - 1)
and
g(x) = lnx
The vertical distance between them at any value of x can be written as
D = f(x) - g(x)
 = Kx + (K - 1) - lnx
We now need to find the minimum value of D and at what value of x it occurs. We find the derivative and equate to zero.
D' = K - 1/x
equating to zero,
K - 1/x = 0
K = 1/x
x = 1/K
Inserting back into the expression for D,
D = K(1/K) + K - 1 - ln(1/K)
 = 1 + K - 1 - ln(K^-1)
 = K - -lnK
 = K + lnK
This is always the minimum distance between the two functions and it clearly depends on K and it increases with increases K. When they almost touch, we could say D = 0 and then
K + lnK = 0
This type of equation can be solved by numerical methods to get K = 0.5671

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Regards