Applications of Derivatives
Function Behavior and Graphing
Grade 12
Question:
<p>For the equation <span class="math">\(\frac{e^{-x}}{1+x} = \lambda\)</span>, which of the following statement(s) is/are correct?</p>
<p>(a) When <span class="math">\(\lambda \in (0, \infty)\)</span>, equation has 2 real and distinct roots</p>
<p>(b) When <span class="math">\(\lambda \in (-\infty, -e^2)\)</span>, equation has 2 real and distinct roots</p>
<p>(c) When <span class="math">\(\lambda \in (0, \infty)\)</span>, equation has 1 real root</p>
<p>(d) When <span class="math">\(\lambda \in (-e, 0)\)</span>, equation has no real root</p>
Step-by-Step Solution
Key Concept: Sketch the graph of the function by finding critical points, asymptotes, and limits, then count intersections with horizontal lines at different heights.
<p>Let <span class="math">$f(x) = \frac{e^{-x}}{1+x}$</span>. Find <span class="math">$f'(x) = \frac{-e^{-x}(1+x) - e^{-x}}{(1+x)^2} = \frac{-e^{-x}(2+x)}{(1+x)^2}$</span>. Critical point at <span class="math">$x = -2$</span>. Analyze the behavior: local maximum at <span class="math">$x = -2$</span> with <span class="math">$f(-2) = e^2$</span>; vertical asymptote at <span class="math">$x = -1$</span>. For different ranges of <span class="math">$\lambda$</span>, determine the number of intersections with horizontal line <span class="math">$y = \lambda$</span>.</p>
Correct Answer: b, c, d