# How do you sketch the graph #y=x^4-x^3-x# using the first and second derivatives?

##### 1 Answer

See below

#### Explanation:

Using the power rule

For turning points of

This is a polynomial of degree 3. To find zeros for polynomials of degree 3 or higher we use Rational Root Test.

The Rational Root Theorem tells you that if the polynomial has a rational zero then it must be a fraction

In this case the factors of the leading coefficient

and factor of the trailing constant

Hence, the possible rational roots of

Testing each in turn reveals

Hence

To find any other real roots:

This is a quadtatic of the form:

Test for real roots:

Hence,

To test the nature of

By inspection it is clear that

To find other zeros:

Unfortunately, this cubic has no rational roots and one real root at

We now have the critical points of

graph{x^4-x^3-x [-10, 10, -5, 5]}

[NB: In practice, it would probably be necessary to plot a few extra points in the interval, say,