Applications of Derivatives
Monotonicity / Injective functions
Grade 12
Question:
<p>If \(f(x) = x^3 + 3x^2 + (4-k)x + b\) is an injective function \(\forall x \in R\), then:</p>
<p>maximum positive integral value of \(k\) is 1.</p>
<p>minimum positive integral value of \(k\) is 1.</p>
<p>number of positive integral value of \(k\) is 1.</p>
<p>number of non-negative integral value of \(k\) is 1.</p>
Step-by-Step Solution
Key Concept: A cubic function is injective (one-to-one) on ℝ if and only if its derivative f'(x) ≥ 0 for all x ∈ ℝ (or f'(x) ≤ 0), meaning the function is monotonic. This requires the discriminant of f'(x) to be non-positive.
<p><strong>Step 1:</strong> Find the derivative: f'(x) = 3x² + 6x + (4-k)</p><p><strong>Step 2:</strong> For f to be injective on ℝ, f'(x) must be non-negative for all x ∈ ℝ (monotonically increasing) or non-positive everywhere (monotonically decreasing).</p><p><strong>Step 3:</strong> Since the coefficient of x² in f'(x) is positive (3 > 0), we need f'(x) ≥ 0 for all x ∈ ℝ.</p><p><strong>Step 4:</strong> This requires the discriminant Δ ≤ 0:</p><p>Δ = 36 - 4(3)(4-k) ≤ 0</p><p>36 - 12(4-k) ≤ 0</p><p>36 - 48 + 12k ≤ 0</p><p>12k - 12 ≤ 0</p><p>k ≤ 1</p><p><strong>Step 5:</strong> Therefore, the function is injective when k ≤ 1. The parameter b can be any real number since it doesn't affect the derivative (only affects vertical translation, which preserves injectivity).</p><p>∴ Answer: k ≤ 1 and b ∈ ℝ (requires specific options to select A,B,C,D)</p>
Correct Answer: A,B,C,D