Question

Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f(x) = x^−6 ln x

Answer 1

First we have to find the derivative of the function:
f ` ( x ) = ( ln x / x^6 ) ` = 
= $\frac{ \frac{1}{x} x^{6}- 6 x^{5}ln x }{ x^{12} } = \\ \frac{ x^{5}(1-6lnx) }{ x^{12} }= \\ \frac{1-6lnx}{ x^{7} }$
The critical point is when f ` ( x ) = 0
1 - 6 ln x = 0
6 ln x = 1
ln x = 1/6
$x = \sqrt[6]{e}$
f ( e^(1/6)) = e^(-1) * 1/6 = 1 / (6 e)
Answer:
 $( \sqrt[6]{e}; \frac{1}{6e} )$