1. y = x^{3}  3x
Find any x and y intercepts, asymptotes, horizontal, vertical, or otherwise, find where the function assumes positive values and where it assumes negative values, the places where the first derivative is either 0 or does not exists, where the function is increasing and decreasing, find the places where the second derivative is 0 or doesn't exist, and where the function is concave up and concave down. Find the local maxima and minima and points of inflection, and sketch the graph.
y  intercept. Let x = 0.
x  intercept. Let y = 0.
We now need to solve for x. To solve a polynomial equation, transpose all terms to one side, leaving a 0 on the other and hope that it factors. Not only do we already have a 0 on the right side, it factors.
Set the factors = 0.


This gives us all the roots or values of x for which f(x) = 0. Since the function is a polynomial and polynomials are continuous functions, the x's for which f(x) = 0 will separate the x's for which f(x) > 0 from the x's for which f(x) < 0. To find where f(x) > 0 and f(x) < 0, will need to evaluate the function at points in between these roots. The points which will be most advantageous to check will be points where f'(x) = 0 or doesn't exist. We find the first derivative
This always exists. The extreme points will then be where the first derivative is 0. We set
This factors
Since 3 is never 0, we can divide both sides by 3 to get
Set the factors = 0


Since
we conclude that
Since
we conclude that
We still need to know what is happening if x^{2} > 3. The simplest such x's to check would be 2 and 2.
so
and
so
Since f(x) is increasing when f'(x) > 0 and decreasing when f'(x) < 0, we use these same techniques on f' to tell when the function is increasing and decreasing. We know when f'(x) = 0, at 1 and 1. We break the real number line into the intevals between these roots. To see whether f'(x) is positive or negative when x < 1, we check and see what happens when x = 2.
so
To see what is happening between 1 and 1, we check out x = 0
so
To see what is happening if x > 1, we check what happens if x = 2
so
At this point we can classify the critical points 1 and 1. f has a relative max when x = 1, because the function stops increasing and starts decreasing. f has a relative min when x = 1 because the function stops decreasing and starts increasing.
Finally we consider concavity and look for points of inflection. For this we need the second derivative.
We quickly and easily see that f"(x) < 0 if x < 0 and f"(x) > 0 if x > 0. Thus f(x) is concave down when x < 0 and concave up when x > 0. Hence there is a point of inflection when x = 0.
We are now ready to sketch the graph. We have plotted the following points
and we can draw the graph.
Polynomials do not have any kind of asymptotes.