Calculus

[Guest]

Given two functions f(x) and g(x) such that their derivative ratio is:
f'(x)/g'(x)=e^f(x)-g(x)
and f(2015)=g(0)=1, find the maximum value of k that f(2015)>k?

[Guest]

Spoiler: show

The question states that [tex]f(2015)=1[/tex], so what you are really asking is what is the maximum value of [tex]k[/tex] such that [tex]1>k[/tex]? The answer is that [tex]k[/tex] has no maximum value. There is no value of [tex]k[/tex] that you could claim is the maximum and satisfies [tex]1>k[/tex] that could not be improved upon by taking [tex](1+k)/2[/tex] instead (this lies strictly between [tex]1[/tex] and [tex]k[/tex] so satisfies the constraint and is bigger than [tex]k[/tex] the supposed maximum).

R. Baber.

[Guest]

Spoiler: show

The question states that [tex]f(2015)=1[/tex], so what you are really asking is what is the maximum value of [tex]k[/tex] such that [tex]1>k[/tex]? The answer is that [tex]k[/tex] has no maximum value. There is no value of [tex]k[/tex] that you could claim is the maximum and satisfies [tex]1>k[/tex] that could not be improved upon by taking [tex](1+k)/2[/tex] instead (this lies strictly between [tex]1[/tex] and [tex]k[/tex] so satisfies the constraint and is bigger than [tex]k[/tex] the supposed maximum).

R. Baber.