Question

Which statements correctly describe the graph of the function f(x) = x3 – 4x2 – 3x + 18? Select three options.

There are three unique solutions to the function when y=0.
The function has a double root.
As x increases from negative infinity to positive infinity, the y-values increase, decrease, and then increase again.
As x approaches negative infinity, y approaches positive infinity.
The domain and range of the function are the set of real numbers.

Answer 1

Correct choices (three):

- The function has a double root.
- As x increases from negative infinity to positive infinity, the y-values increase, decrease, and then increase again.
- The domain and range of the function are the set of real numbers.

Brief justification:
- f(x) = x^3 − 4x^2 − 3x + 18 = (x − 3)^2(x + 2), so roots are x = 3 (double) and x = −2 (simple) — not three distinct zeros.
- f'(x) = 3x^2 − 8x − 3 = 3(x − 3)(x + 1/3), so critical points at x = −1/3 (local max) and x = 3 (local min). Hence the graph increases, then decreases, then increases.
- As a cubic with positive leading coefficient it ranges over all real numbers and its domain is all real numbers; also as x → −∞, f(x) → −∞ (not +∞).