MICHAEL S. answered • 10/15/19

SCIENCE AND MATH HELP

f(x) = x ^{-6 ln(x)}

One of the easiest ways to do this is by logarithmic differentiation. I'm going to use y for f(x).

__Step 1:__ First, take the natural log of both sides.

ln y = ln [ x ^{-6 ln(x)}]

__Step 2:__ Using one of the log rules (ln b^{c} = c ln b)

ln y = -6 ln(x) ln (x)

__Step 3:__ Now, differentiate both sides. On the right side, use the product rule.On the left side, use implicit differentiation.

[1 / y] dy/dx = -6 [ln(x) (1/x) + (1/x) ln(x)]

__Step 4:__ Simplify.

[1 / y] dy/dx = -6 [2 ln(x) / x]

= - 12 ln(x) / x

__Step 5__: Multiply both sides by y.

dy/dx = - 12 y ln(x) / x

__Step 6:__ Substitute the expression for y given in the problem.(y = f(x) = x ^{-6 ln(x)})

dy/dx = - 12 x ^{-6 ln(x)} ln(x) / x

__Step 7:__ The critical numbers are where dy/dx = 0

dy/dx = - 12 x ^{-6 ln(x)} ln(x) / x = 0

__Step 8:__ Consider the domain of the function.

One of the restrictions of the natural log function is that 0 is not part of its domain. So x cannot equal zero. There is only one value that makes dy/dx = 0. It is the value at which...

ln (c) = 0, where c is the critical number.

That value of c is 1. (c = 1)

__Final Answer:__ Critical number x = 1.