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,
