How can write the following function ?

Questions
1

{f[x] == (a x^3 + 6 x^2 + b)/(c x^2) with a?0 and b?0, Maximum relative (-2,0), oblique asymptotes ( y=2/3x + 2 )}
Thanks
Mike

2

In Mathematica execute

Maximize[ {(a x^3 + 6 x^2 + b)/(c x^2), -2 < x < 0} , x]

to get the full answer qualified by constraints on the parameters.

3

In[4]:= y[x_] := (a x^3 + 6 x^2 + b)/(c x^2)

In[5]:= y[x]

Out[5]= (b + 6 x^2 + a x^3)/(c x^2)

In[6]:= Expand[%]

Out[6]= 6/c + b/(c x^2) + (a x)/c

As x->Infinity, asymptotic form implies c=3, a=2. Don’t understand " Maximum relative (-2,0)"

4

local maximum (relative maximum): a value of a function that is greater than those values of the function at the surrounding points, but is not the greatest of all values.

5

How can qualified in Mathematica the constraints to get the full answer?
Thanks a lot

6

I see. Continuing from above

In[3]:= y[x_] := (2 x^3 + 6 x^2 + b)/(3 x^2)

In[4]:= y'[-2]

Out[4]= (8 + b)/12

(* y'[2]=0 gives b=-8 *)

In[5]:= Plot[(2 x^3 + 6 x^2 - 8)/(3 x^2), {x, -4, 4}]

This gives your function