Vertical asymtotes,domain, c of natural log function

Question

Hi, I need to sketch the graph for y=x(4-x)-3lnx . It asks for the domain,x and y intercepts, asymptotes, 1st and second derivative to find critical points. 

 

How do I find all of these. The Ln is confusing me

Answer 1

You need to first find critical information in order to graph it.  Since your function is a bit unclear to me, I will give you guidelines to follow.

 

To find the x-intercepts, set y=0 and solve for x.

 

To find the y-intercepts, set x=0 and solve for y.

 

To find the domain, we examine any terms in the function that may cause some restrictions for x.  Look at the term lnx.  Since you cannot have a negative value for logarithms, your domain is   x>0.  The domain will also affect the x-intercept that you found earlier.

 

The vertical asymptote is  x=0

 

 

Now we find the critical points.  To find the critical points, set the derivative of y equal to zero and solve for x.  The x values indicates the possible location of the critical points.  You will then evaluate the derivative of the function around these x values.  If the derivative changes from positive to negative, then you have a maximum value.  Otherwise, minimum value.

 

To find the points of inflection, set the second derivative of y equal to zero and solve for x.  Second derivative means to take the derivative of the first derivative.  The x values that you obtain will be your possible points of inflection. You will then evaluate the second derivative of the function around these x values.  If the second derivative changes signs around these x values, then you have a point of inflection.  If the second derivative is positive, then concave up.  Otherwise concave down.