How do you sketch the graph of #f(x)=x^33x^2#?
2 Answers
Calculate derivative :
So you can study the sign of
#f'(x) < 0 iff x in ]0,2[# #f'(x) > 0 iff x in ]oo, 0[ cup ]2,+oo[# .
You get now the variations of


Remark that
 local maximum in 0 with
 local minimum in 2 with
There are two horizontal tangents at 0 and at 2.
 For the limits, apply the rule :
#lim_{x>oo} x^33x^2 = lim_{x>oo} x^3 = oo#
#lim_{x>+oo} x^33x^2 = lim_{x>+oo} x^3 = +oo#
Finally :
graph{x^3  3x^2 [8.42, 13.78, 6.62, 4.48]}
You can start by setting
Setting
When
Points of maximum or minimum are found by setting the first derivative equal to zero:
and
and
When
Setting the second derivative equal to zero will give us inflection point(s):
and
so that
And finally: