Question

Which is the graph of the function f(x) = x3 + x2 + 4?

Answer 1

The graph is a cubic with positive leading coefficient, so it falls to the left and rises to the right. Key features:

- f(x) = x^3 + x^2 + 4 = (x + 2)(x^2 − x + 2), so the only real x-intercept is x = −2 (the quadratic factor has no real roots).
- y-intercept: f(0) = 4, so it passes through (0, 4).
- Critical points from f'(x) = 3x^2 + 2x = x(3x + 2): x = 0 and x = −2/3.
- f(0) = 4 is a local minimum (f''(0) = 2 > 0).
- f(−2/3) = 112/27 ≈ 4.148 is a local maximum (f''(−2/3) = −2 < 0).
- Behavior: comes from −∞ at left, crosses the x-axis at (−2,0), rises to a small local maximum near (−2/3, 4.148), dips slightly to a local minimum at (0,4), then increases to +∞ as x → +∞.

So look for the cubic that crosses the x-axis at −2, has y-intercept 4, and a tiny “wiggle” near y ≈ 4 between x = −2/3 and x = 0 (local max then local min).