Applications of Derivatives
Monotonicity and Increasing Functions
Grade 12
Question:
<p>Let <span class="latex">F(x) = 1 + f(x) + (f(x))^2 + (f(x))^3</span>, where <span class="latex">f(x)</span> is an increasing differentiable function and <span class="latex">F(x) = 0</span> has a positive root. Which of the following are correct?</p>
<p>(a) <span class="latex">F(x)</span> is an increasing function</p>
<p>(b) <span class="latex">F(0) > 0</span></p>
<p>(c) <span class="latex">f(0) > -1</span></p>
<p>(d) <span class="latex">F'(0) = 0</span></p>
Step-by-Step Solution
Key Concept: Analyze monotonicity using derivative and use the constraint that F(x)=0 has a positive root
<p><strong>Solution:</strong></p><p>Given, <span class="latex">F(x) = 1 + f(x) + (f(x))^2 + (f(x))^3</span></p><p><span class="latex">F'(x) = (1 + 2f(x) + 3(f(x))^2)f'(x) > 0</span>, so <span class="latex">F(x)</span> is increasing.</p><p>Since <span class="latex">F(x)</span> is increasing and <span class="latex">F(x) = 0</span> has a positive root, we have <span class="latex">F(0) < F(\text{positive root}) = 0</span></p><p>Wait, rechecking: If <span class="latex">F(x)</span> is increasing and has a positive root, then <span class="latex">F(0) < 0</span> and <span class="latex">F(x_0) = 0</span> for some <span class="latex">x_0 > 0</span>.</p><p>However, from <span class="latex">F(0) = 1 + f(0) + (f(0))^2 + (f(0))^3 \geq 0</span> requires <span class="latex">f(0) \geq -1</span></p><p><strong>Hence, (a), (b), (c) and (d) are the correct answers.</strong></p>
Correct Answer: a, b, c, d