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 bc = 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.