Question

A function is shown below: (x) = −x^3 - 5x^2 + 9x + 45. Circle the letters of all the statements that are true.

a) As x approaches negative infinity, y approaches negative infinity.
b) As x approaches positive infinity, y approaches negative infinity.
c) The function is decreasing where x is less than –4.07, and where x is greater than 0.74.
d) The function has a relative minimum at about (–4.07, –7.04).

Answer 1

The given function has certain properties as x approaches infinity and negative infinity, and specific critical points. It is true that as x approaches negative infinity and positive infinity, y approaches negative infinity. The function is also decreasing in certain intervals and has a relative minimum at approximately (-4.07, -7.04).

Step-by-step explanation:

To determine which statements are true about the given function, we can analyze its behavior as x approaches infinity and negative infinity, and find the critical points where the function is increasing or decreasing.

a) As x approaches negative infinity, the function approaches negative infinity. This is true because the leading term of the function is -x^3, which dominates the other terms as x approaches negative infinity.

b) As x approaches positive infinity, the function approaches negative infinity. This is true because the leading term of the function is -x^3, which dominates the other terms as x approaches positive infinity.

c) The function is decreasing where x is less than –4.07, and where x is greater than 0.74. This is true because the function has a relative maximum at x ≈ -4.07 and a relative minimum at x ≈ 0.74.

d) The function has a relative minimum at about (–4.07, –7.04). This is true because the function has a relative minimum at x ≈ -4.07, and the corresponding y-value is approximately -7.04.