Applications of Derivatives
Stationary Points
Grade 12
Question:
<p>The set of all values of <span class="math">b</span> for which the function <span class="math">f(x) = (b^2 + 3b + 2)(\cos^2 x + \sin^2 x) + (b + 1)x + \sin 2</span> does not possess stationary points is</p>
<p>(a) <span class="math">[1, \infty)</span></p>
<p>(b) <span class="math">(0, 1) \cup (1, 4)</span></p>
<p>(c) <span class="math">\left[\frac{3}{2}, \frac{5}{2}\right]</span></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Since the function simplifies to a linear function in x (constant coefficient + bx), f'(x) is just the slope, which must be non-zero for no stationary points.
<p><strong>Solution:</strong></p><p>Given: <span class="math">f(x) = (b^2 + 3b + 2)(\cos^2 x + \sin^2 x) + (b + 1)x + \sin 2</span></p><p>Since <span class="math">\cos^2 x + \sin^2 x = 1</span>:</p><p><span class="math">f(x) = (b^2 + 3b + 2) + (b + 1)x + \sin 2</span></p><p><span class="math">f'(x) = (b + 1)</span></p><p>For no stationary points: <span class="math">f'(x) \neq 0</span> for all <span class="math">x</span></p><p>Therefore: <span class="math">b + 1 \neq 0</span></p><p><span class="math">b \neq -1</span></p><p>The answer is <strong>(d) None of these</strong>.</p>
Correct Answer: d