Select all the correct locations on the image.

Consider the function [tex]f[/tex].
[tex]\[f(x) = x^3 + 2x^2 - 5x - 6\][/tex]

1. Select the locations of the zeros of function [tex]f[/tex] on the coordinate plane.
2. Select the end behavior of its graph:
- As [tex]x[/tex] approaches positive infinity, [tex]f(x)[/tex] approaches negative infinity.
- As [tex]x[/tex] approaches negative infinity, [tex]f(x)[/tex] approaches positive infinity.
- As [tex]x[/tex] approaches positive infinity, [tex]f(x)[/tex] remains constant.
- As [tex]x[/tex] approaches negative infinity, [tex]f(x)[/tex] approaches negative infinity.



Answer :

Let's break down the information given to find the zeros of the function [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex] and analyze its end behavior.

### Finding the zeros of the function

To find the zeros of [tex]\( f(x) \)[/tex], we need to solve the equation:
[tex]\[ x^3 + 2x^2 - 5x - 6 = 0 \][/tex]

Numerical solutions of this equation are:
[tex]\[ x_1 \approx -3, \quad x_2 \approx -1, \quad x_3 \approx 2 \][/tex]

These are the points where the function [tex]\( f(x) \)[/tex] crosses the x-axis.

### Analyzing the end behavior

To understand the end behavior of the function, consider the leading term of the polynomial, which is [tex]\( x^3 \)[/tex]. The leading term primarily determines the end behavior of the polynomial function.

For the cubic function [tex]\( f(x) = x^3 + 2x^2 - 5x - 6 \)[/tex]:

1. As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to +\infty \)[/tex]), the value of [tex]\( f(x) \)[/tex] will be dominated by the leading term [tex]\( x^3 \)[/tex], which also approaches positive infinity because a positive number raised to an odd power (like 3) remains positive and grows very large.
[tex]\[ \lim_{x \to +\infty} f(x) = +\infty \][/tex]

2. As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), the value of [tex]\( f(x) \)[/tex] will also be dominated by the leading term [tex]\( x^3 \)[/tex], which approaches negative infinity because a negative number raised to an odd power (like 3) remains negative and grows very large in the negative direction.
[tex]\[ \lim_{x \to -\infty} f(x) = -\infty \][/tex]

### End behavior selection

From the analysis above, we can conclude the following:

- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.

- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.

Therefore, the correct end behavior is:

- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.

### Final selected points and end behavior

- The zeros of the function [tex]\( f(x) \)[/tex] are approximately at [tex]\( x = -3 \)[/tex], [tex]\( x = -1 \)[/tex], and [tex]\( x = 2 \)[/tex].
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.