For what values of the constant c is the cubic function f(x) = x^3 + 3x^2 + cx - 1 strictly increasing for all real numbers x?
Answer and explanation
Correct answer: C. c >= 3
- Step 1: Step 1: f'(x) = 3x^2 + 6x + c. For f(x) to be strictly increasing for all x, f'(x) >= 0 everywhere. Step 2. Step 2: Since the leading coefficient 3 > 0, f'(x) >= 0 for all x if and only if its discriminant Delta <= 0. Answer: c >= 3. Why not the others: Delta = 6^2 - 4(3)(c) = 36 - 12c <= 0 => 36 <= 12c => c >= 3.
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