Applications of Derivatives
Properties of Functions and Inverses
Grade 12
Question:
<p>If <span class="math">f : \mathbb{R} \to \mathbb{R}</span>, <span class="math">f(x)</span> is a differentiable bijective function, then which of the following is true?</p>
<p>(a) <span class="math">(f(x) - x)f''(x) > 0, \forall x \in \mathbb{R}</span></p>
<p>(b) <span class="math">(f(x) - x)f''(x) \leq 0, \forall x \in \mathbb{R}</span></p>
<p>(c) If <span class="math">(f(x) - x)f''(x) \leq 0</span>, then <span class="math">f(x) = f^{-1}(x)</span> has no solution</p>
<p>(d) If <span class="math">(f(x) - x)f''(x) \leq 0</span>, then <span class="math">f(x) = f^{-1}(x)</span> has at least one real solution</p>
Step-by-Step Solution
Key Concept: Analyze the conditions on the product (f(x) - x)f''(x) to determine properties of bijective functions and their inverses.
<p><strong>Solution:</strong></p><p>Since <span class="math">(f(x) - x)f''(x) > 0</span> for all <span class="math">x \in \mathbb{R}</span>, either:</p><p>Case 1: <span class="math">(f(x) - x) \geq 0</span> and <span class="math">f''(x) > 0</span></p><p>Case 2: <span class="math">(f(x) - x) > 0</span> and <span class="math">f''(x) \leq 0</span></p><p>Case 1 is not possible if <span class="math">f(x) - x = 0</span> and <span class="math">f'(x)</span> are decreasing, then <span class="math">f(x)</span> has to intersect the line <span class="math">y = x</span>.</p><p>Similarly, if <span class="math">f(x) - x > 0</span> and <span class="math">f''(x) \leq 0</span>, this is not possible.</p><p>Also, if <span class="math">f(x) - x \leq 0</span>, then <span class="math">f(x) = f^{-1}(x)</span> has no solution.</p><p>∴ Answer is (c).</p>
Correct Answer: D