Calculus need help please?

Question

4. Determine if  is increasing or decreasing at the following points.

(a) z=1 

(b) z=5 

(c) z=20

Answer 1

Take the derivative. derivative of a product is the 1st time times derivative of the 2nd term plus the 2nd term times derivative of the 1st term

G' = (z-6)(lnz)' + lnz(z-6)'

G' = (z-6)(1/z) + lnz(1) = 1 -6/z + lnz

G'(1) = 1-6+ 0 = -5. It's decreasing. Negative slope means it's decreasing

G'(5) = 1-6/5 +ln5 = -1/5 + 1.6 = +1.4 It's increasing

G'(20) = 1-6/20 + ln20 = .7+ 3 = +3.7. It's increasing