Question

please can anyone summarize the process of drawing the graphs of a curve, by using calculous? ie, by finding retricemptotes and extremum.

Answer 1

1) Find out the function's domain;
2)Do f'(x), then see when f'(x)>0, f'(x)<0, and when f'(x)=0;
3)Do f''(x), then find out when f''(x)>0, and when f''(x)<0;
4)Do the limits of the function when x goes to infinity, and - infinity;
5)Find out the function's roots.
Points where f'(x)=0 are maximum or minimum. For example, f'(b)=0,f'(b-1)<0, and f'(b+1)>0 => (x,f(x)) is a minimum.
If f''(x)>0 the function is concave up in this point; if f''(x)<0 the function is concave down in this point. If f''(x)=0, x could be a inflection point. To test that, do the same as the last step, see if f''(x) changes its signal before and after x.
Its also very good to do the limits of the function in the point that are not in the functions domain and when x goes to zero.
I'm sorry, my english is no that good, but I hope you understand.

Answer 2

it is fairly simple first find dy/dx pr f'(x)
equate (dy/dx)=0 or f'(x)=0 and solve it
This would give you some value
now again differentiate (dy/dx) or find f''(x)
check whether f''(x) at the values which you obtained by equating f'(x)=0
if f''(x)<0 then it is a point of maxima
if f''(x)>0 then it is point of minima
if f''(x)=0 then it is a stationary point or a point of inflexion and function has no maxima or minima

P.S dy/dx and f'(x) are one and the same thing