Let
𝑓
(
𝑥
)
=
𝑥
3
+
𝑥
2
−
2
𝑥
+
3
. Which of the following is the set on which
𝑓
is increasing.

Question

Let 
𝑓
(
𝑥
)
=
𝑥
3
+
𝑥
2
−
2
𝑥
+
3
. Which of the following is the set on which 
𝑓
 is increasing.

BlackTom AI · Accepted Answer

f'(x) = 3x^2 + 2x − 2, and monotonicity of a differentiable function is determined by the sign of its derivative. 1) Relevant concepts - A function is increasing where its derivative is positive. - For a quadratic derivative ax^2+bx+c with a>0, the quadratic is positive outside the interval between its real roots and negative between them. 2) Solve f'(x) > 0 step by step Compute the derivative: f'(x) = 3x^2 + 2x − 2. Find the roots of f'(x) = 0 using the quadratic formula: Discriminant Δ = b^2 − 4ac = 2^2 − 4(3)(−2) = 4 + 24 = 28. Roots: x = [−2 ± √28]/(2·3) = [−2 ± 2√7]/6 = [−1 ± √7]/3. So the two critical points are x1 = (−1 − √7)/3 and x2 = (−1 + √7)/3. 3) Sign analysis (intermediate reasoning) - The coefficient of x^2 in f'(x) is 3 (>0), so the parabola opens upward. - Therefore f'(x) > 0 for x < x1 and for x > x2, and f'(x) < 0 for x1 < x < x2. 4) Why the correct answer is correct Because f' is positive outside the interval between the two real roots, f is increasing on the union of the two exterior intervals: (−∞, (−1 − √7)/3) ∪ ((−1 + √7)/3, ∞). 5) Why other possibilities are incorrect Any interval that lies between the two roots would have f'(x) < 0, so f would be decreasing there; any single interval that does not match the exterior intervals would fail to capture all x where f'(x)>0. Conclusion The set on which f is increasing is (−∞, (−1 − √7)/3) ∪ ((−1 + √7)/3, ∞), which matches the provided answer.

Let 𝑓 ( 𝑥 ) = 𝑥 3 + 𝑥 2 − 2 𝑥 + 3 . Which of the following is the set on which 𝑓 is increasing.单项选择题

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