Applications of Derivatives
Critical Points and Extrema
Grade 12
Question:
<p>If <span class="math">\((f(x) - 1)(x^2 + x + 1)^2 - (f(x) + 1)(x^4 + x^2 + 1) = 0\)</span> for all <span class="math">\(x \in \mathbb{R} - \{0\}\)</span> and <span class="math">\(f(x) \neq \pm 1\)</span>, then which of the following statement(s) is/are correct?</p>
<p>(a) <span class="math">\(|f(x)| \geq 2\)</span> for all <span class="math">\(x \in \mathbb{R} - \{0\}\)</span></p>
<p>(b) <span class="math">\(f(x)\)</span> has a local maximum at <span class="math">\(x = -1\)</span></p>
<p>(c) <span class="math">\(f(x)\)</span> has a local minimum at <span class="math">\(x = 1\)</span></p>
<p>(d) <span class="math">\(-\int_{-\pi}^{\pi} (\cos x) f(x) \, dx = 0\)</span></p>
Step-by-Step Solution
Key Concept: Solve the functional equation to find an explicit form of f(x), then use calculus and algebra to verify all statements.
<p>Rearrange the equation to solve for <span class="math">\(f(x)\)</span>: <span class="math">\(f(x) = \frac{(x^2+x+1)^2 + (x^4+x^2+1)}{(x^2+x+1)^2 - (x^4+x^2+1)}\)</span>. Simplify the denominator using <span class="math">\((x^2+x+1)^2 = x^4 + 2x^3 + 3x^2 + 2x + 1\)</span> and <span class="math">\(x^4+x^2+1\)</span>, giving <span class="math">\(2x^3 + 2x^2 + 2x = 2x(x^2+x+1)\)</span>. Thus <span class="math">\(f(x) = \frac{2x^4 + 2x^3 + 4x^2 + 2x + 2}{2x(x^2+x+1)} = \frac{x^4+x^3+2x^2+x+1}{x(x^2+x+1)}\)</span>. After simplification, <span class="math">\(f(x) = x + \frac{1}{x} + 1\)</span>. Then <span class="math">|f(x)| = |x + \frac{1}{x} + 1| \geq 2 + 1 = 3\)</span> by AM-GM (or verify directly). <span class="math">f'(x) = 1 - \frac{1}{x^2} = 0\)</span> at <span class="math">x = \pm 1\)</span>. At <span class="math">x = -1\)</span>, <span class="math">f(-1) = -1 - 1 + 1 = -1\)</span> (local max); at <span class="math">x = 1\)</span>, <span class="math">f(1) = 1 + 1 + 1 = 3\)</span> (local min). For the integral, note that <span class="math">\(\cos x\)</span> is even and <span class="math">f(x)\)</span> is odd in structure or use direct calculation.</p>
Correct Answer: a, b, c, d