Increasing/behavior behavior of derivative

Question

y= 4x-x^2-3lnx 

 

f'(x)= 4-2x-(3/x) 

 

My book is saying that its always decreasing, but when i look at the behavior before 0 , its negative and then after 0 its positive. Im confused

Answer 1

f(x) = 4x - x² - 3lnx

 

f'(x) = 4 - 2x -3/x

 

       = (-2x² -4x -3)/x = -(2x²+4x+3)/x

 

lnx is defined only when x>0. So, the domain of f(x) is (0,∞).

 

When x > 0, 2x²+4x+3 > 0

 

So, f'(x) < 0 for all x>0.  Therefore, f(x) is decreasing on (0, ∞).