Find the Behavior (Leading Coefficient Test) f(x)=4x^3-4x-x^5

$f (x) = 4 x^{3} - 4 x - x^{5}$

Step 1

Identify the degree of the function.

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$4 x^{3} \to 3$

$- 4 x \to 1$

$- x^{5} \to 5$

The largest exponent is the degree of the polynomial.

$5$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Move $- 4 x$ .

$4 x^{3} - x^{5} - 4 x$

Reorder $4 x^{3}$ and $- x^{5}$ .

$- x^{5} + 4 x^{3} - 4 x$

The leading term in a polynomial is the term with the highest degree.

$- x^{5}$

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 1$

Step 4

Since the leading coefficient is negative, the graph falls to the right.

Negative

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Rises to the left and falls to the right

Step 7