Applications of Derivatives
Monotonicity
Grade 12
Question:
<p>Given function is \(f(x) = \cot^{-1}x + x\); \(D_f = \mathbb{R}\). Which of the following is true?</p>
<p>\(f(x)\) is increasing \(\forall\, x \in (-\infty, \infty)\)</p>
<p>\(f(x)\) is decreasing \(\forall\, x \in (-\infty, \infty)\)</p>
<p>\(f(x)\) is neither increasing nor decreasing</p>
<p>\(f(x)\) is increasing in \((0, \infty)\) only</p>
Step-by-Step Solution
Key Concept: To determine if f(x) = cot⁻¹(x) + x is monotonic, find f'(x) and check its sign across ℝ. The derivative of cot⁻¹(x) is -1/(1+x²), which is always negative, but we must evaluate the net derivative.
<p><strong>Step 1:</strong> Find f'(x) for f(x) = cot⁻¹(x) + x</p><p>f'(x) = d/dx[cot⁻¹(x)] + d/dx[x] = -1/(1+x²) + 1</p><p><strong>Step 2:</strong> Simplify f'(x)</p><p>f'(x) = -1/(1+x²) + 1 = [-(1) + (1+x²)]/(1+x²) = x²/(1+x²)</p><p><strong>Step 3:</strong> Analyze the sign of f'(x)</p><p>Since x² ≥ 0 and 1+x² > 0 for all x ∈ ℝ, we have f'(x) ≥ 0 for all x ∈ ℝ</p><p>f'(x) = 0 only at x = 0; f'(x) > 0 for x ≠ 0</p><p><strong>Step 4:</strong> Conclusion</p><p>Since f'(x) ≥ 0 throughout ℝ and is strictly positive except at one isolated point, f(x) is strictly increasing on ℝ.</p><p>∴ Answer: A (f is strictly increasing on ℝ)</p>
Correct Answer: A