How do you find the relative extrema for f(x) = x - log_4xf(x)=xlog4x?

1 Answer
Apr 26, 2016

The minimum of f(x)f(x) is f(1/(ln 4))=0.957f(1ln4)=0.957, nearly.

Explanation:

Use log_b a = log_c a log_a clogba=logcalogac

ln x=log_e x=log_4 x log_e 4=log_4 x ln 4lnx=logex=log4xloge4=log4xln4.

log_4 x=ln x/ln 4log4x=lnxln4.

Now, f(x)=x-ln x/ln 4f(x)=xlnxln4

f'=1-1/(x ln 4)=0, when x=1/ln 4

f''=1/(x^2 ln 4)>0, for all x, as ln 4=1.3863>0..

Thus, f(1/ln 4) is the minimum of f(x).

f(1/ln 4)=1/ln 4-ln(1/ln 4)/ln 4=(1+ln(ln 4))/ln 4=0.957, nearly.-