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Which is the graph of the function f(x) = x3 + x2 + x + 1?
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GPT-5 mini
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Factor and analyze:
f(x) = x^3 + x^2 + x + 1 = (x+1)(x^2+1).
Key features:
- Real root: x = −1 (x^2+1 has no real roots), so the graph crosses the x-axis once at (−1,0).
- y-intercept: f(0) = 1, so it passes through (0,1).
- Leading coefficient positive (1) → as x→∞, f→∞; as x→−∞, f→−∞.
- Derivative f'(x) = 3x^2 + 2x + 1 has discriminant 4−12 < 0, so f'(x) > 0 for all x: the function is strictly increasing (no local maxima or minima).
- Inflection point where f''(x)=6x+2=0 → x = −1/3; f(−1/3) = 20/27 ≈ 0.7407.
Thus the correct graph is a smooth cubic increasing everywhere, crossing the x-axis at (−1,0), passing through (0,1), with an inflection near (−1/3, 0.7407) and no turning points. Choose the curve with those properties.
f(x) = x^3 + x^2 + x + 1 = (x+1)(x^2+1).
Key features:
- Real root: x = −1 (x^2+1 has no real roots), so the graph crosses the x-axis once at (−1,0).
- y-intercept: f(0) = 1, so it passes through (0,1).
- Leading coefficient positive (1) → as x→∞, f→∞; as x→−∞, f→−∞.
- Derivative f'(x) = 3x^2 + 2x + 1 has discriminant 4−12 < 0, so f'(x) > 0 for all x: the function is strictly increasing (no local maxima or minima).
- Inflection point where f''(x)=6x+2=0 → x = −1/3; f(−1/3) = 20/27 ≈ 0.7407.
Thus the correct graph is a smooth cubic increasing everywhere, crossing the x-axis at (−1,0), passing through (0,1), with an inflection near (−1/3, 0.7407) and no turning points. Choose the curve with those properties.
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