Find constants a and b in the function f(x)=ax^b/(ln(x)) such that f(19)=1 and the function has a local minimum at x=19. What is a and b?

Answer 1

Just plug and chug:

f = ax^b/lnx
1 = a*19^b / ln 19

a * 19^b = 2.944

f' = (abx^(b-1) * lnx) - ax^(b-1))/ln^2(x)
= [ax^(b-1) * (b*ln x-1)]/ln^2(x)

to get f'=0, we need b*lnx = 1

b*ln19 = 1
b = 1/ln19

a* 19^b = 2.944
a*19^(1/ln19) = a * 19^.3396 = a*2.718 = 2.944
a = 1.083

So, if my math is right, f(x) = 1.083*x^.3396/lnx