Applications of Derivatives
Increasing and Decreasing Functions
Grade 12
Question:
<p>Let \(f : [1, \infty) \to \mathbb{R}\) and \(f(x) = x-1\). Then which statement is true?</p><p>(A) f(x) is an increasing function</p><p>(B) \(\lim_{x \to \infty} f(x) = \infty\)</p><p>(C) f(x) has a maxima at \(x = e\)</p><p>(D) f(x) is a decreasing function</p>
<p>(A) f(x) is an increasing function</p>
<p>(B) \(\lim_{x \to \infty} f(x) = \infty\)</p>
<p>(C) f(x) has a maxima at \(x = e\)</p>
<p>(D) f(x) is a decreasing function</p>
Step-by-Step Solution
Key Concept: A function is increasing when its derivative is positive. Linear functions with positive slope are strictly increasing.
<p><strong>Step 1:</strong> For $f(x) = x - 1$ on $[1, \infty)$, the derivative is $f'(x) = 1 > 0$.</p><p><strong>Step 2:</strong> Since the derivative is positive, f(x) is increasing.</p><p><strong>Step 3:</strong> As $x \to \infty$, $f(x) \to \infty$, which is also true.</p><p>∴ Both (A) and (B) are correct, but (A) is the primary characterization.</p>
Correct Answer: A