Question

for the polynomial function \\(f(x) = 4x^3 - 4x - x^5\\), answer the parts a through e.

a. the graph of \\(f(x)\\) falls to the left and rises to the right.
b. the graph of \\(f(x)\\) rises to the left and falls to the right.
c. the graph of \\(f(x)\\) falls to the left and falls to the right.
d. the graph of \\(f(x)\\) rises to the left and rises to the right.

b. find the x-intercept(s). state whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
the x-intercept(s) is/are \\(0, \sqrt{2}, -\sqrt{2}\\)
(type an exact answer, using radicals as needed. use a comma to separate answers as needed. type each answer only once.)

at which x-intercept(s) does the graph cross the x-axis? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x-intercept(s) at which the graph crosses the x-axis is/are \\(0\\)
(type an exact answer, using radicals as needed. use a comma to separate answers as needed. type each answer only once.)
b. there are no x-intercepts at which the graph crosses the x-axis.

at which x-intercept(s) does the graph touch the x-axis and turn around? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x-intercept(s) at which the graph touches the x-axis and turns around is/are \\(\sqrt{2}, -\sqrt{2}\\)
(type an exact answer, using radicals as needed. use a comma to separate answers as needed. type each answer only once.)
b. there are no x-intercepts at which the graph touches the x-axis and turns around.

c. find the y-intercept.

Explanation:

Identify the polynomial function and its degree

$$ f(x) = -x^5 + 4x^3 - 4x $$

The leading term is \(-x^5\). The degree is \(5\) (odd) and the leading coefficient is \(-1\) (negative).

Determine the end behavior

Since the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right.

Find the y-intercept

Evaluate the function at \(x = 0\):

$$ f(0) = -0^5 + 4(0)^3 - 4(0) = 0 $$

Answer:

The y-intercept is 0.