Question

Which is the graph of the function f(x) = x3 + 6x2 + 11x + 6?

Answer 1

First factor it:
f(x) = x^3 + 6x^2 + 11x + 6 = (x+1)(x+2)(x+3).

Key features:
- x‑intercepts (real roots) at x = −1, −2, −3.
- y‑intercept f(0) = 6.
- Leading coefficient > 0, so as x → +∞, f(x) → +∞; as x → −∞, f(x) → −∞.
- Critical points from f'(x) = 3x^2 + 12x + 11 = 0 give
x = (-12 ± √12)/6 = -2 ± (√3)/3 ≈ −2.577 and −1.423.
f has a local maximum ≈ 0.385 at x ≈ −2.577 and a local minimum ≈ −0.385 at x ≈ −1.423.

So the graph is a cubic that comes from −∞ on the left, rises to a small positive local maximum between x = −3 and −2, falls crossing the x-axis at −2 and −1 to a small negative local minimum, then rises through (0,6) to +∞ on the right. Choose the graph showing roots at −3, −2, −1 and that shape.