Question

Which is the graph of the function f(x) = 2x3 – 7x2 + 2x + 3?

Answer 1

Assuming you mean f(x) = 2x^3 − 7x^2 + 2x + 3, useful facts to identify its graph:

- Factorization / x-intercepts: f(x) = (x − 1)(2x + 1)(x − 3).
So zeros at x = −1/2, 1, 3 (all simple roots, the curve crosses the axis at each).

- End behavior: leading coefficient > 0 and degree 3 ⇒ as x → −∞, f(x) → −∞; as x → +∞, f(x) → +∞.

- y-intercept: f(0) = 3.

- Critical points (local extrema): f'(x) = 6x^2 − 14x + 2 = 0 ⇒ x ≈ 0.1529 (local max) and x ≈ 2.1805 (local min).
f(0.1529) ≈ 3.149, f(2.1805) ≈ −5.183.

- Inflection point: f''(x) = 12x − 14 = 0 ⇒ x = 7/6 ≈ 1.1667, f(7/6) ≈ −1.0185.

So the correct graph: a cubic that goes down on the far left, crosses the x-axis at x = −0.5, rises to a small local maximum just above y = 3 near x ≈ 0.153, then falls crossing the x-axis at x = 1, drops to a deep local minimum near (2.18, −5.18), then rises, crossing the x-axis at x = 3 and continuing upward. Choose the graph with these features.