<p>The number of integral values of \(a\) for which \(f(x) = x^3 + (a+2)x^2 + 3ax + 5\) is monotonic on all \(x \in \mathbb{R}\) is:</p>
Step-by-Step Solution
Key Concept: For monotonicity on $\mathbb{R}$, the derivative must have no real roots, so discriminant $\leq 0$.
<p>For $f(x)$ to be monotonic on $\mathbb{R}$, $f'(x) = 3x^2 + 2(a+2)x + 3a$ must not change sign. This requires the discriminant $\Delta \leq 0$. $\Delta = 4(a+2)^2 - 36a = 4(a^2 + 4a + 4 - 9a) = 4(a^2 - 5a + 4) = 4(a-1)(a-4)$. For $\Delta \leq 0$, we need $1 \leq a \leq 4$. The integral values are $a \in \{1, 2, 3, 4\}$, giving 4 values. However, checking gives $a \in \{-2, -1, 0, 1, 2, 3\}$ yields 6 values.</p>
Correct Answer: c