Applications of Derivatives
Increasing and Decreasing Functions
Grade 12
Question:
<p>The function <span>\(f(x) = \cot^{-1} x + x\)</span> increases in the interval</p>
<p>(a) \((1, \infty)\)</p>
<p>(b) \((-1, \infty)\)</p>
<p>(c) \((-\infty, \infty)\)</p>
<p>(d) \((0, \infty)\)</p>
Step-by-Step Solution
Key Concept: A function is increasing when its derivative is non-negative throughout the domain. Calculate the derivative and check its sign.
<p><strong>Solution:</strong></p><p>Since $f(x) = \cot^{-1} x + x$</p><p>On differentiating w.r.t. $x$, we get:</p><p>$f'(x) = -\frac{1}{1+x^2} + 1 = \frac{x^2}{1+x^2} \geq 0$</p><p>Hence, $f(x)$ is increasing function for all $x \in (-\infty, \infty)$.</p><p>∴ Answer is (c).</p>
Correct Answer: C