Let f: \mathbb{R} \to \mathbb{R} and f(x) = x^3 + ax^2 + bx - 8. If f(x) = 0 has three real roots & f(x) is a bijective function, then (a + b) is equal to
Step-by-Step Solution
<div class="solution"><p>Shift relation by ±1/2: combine to get f(x+1)+f(x)+f(x-1)=0.</p><p>This standard recurrence gives f(x+3)=f(x), so period=3.</p><p>Greater than 1✓, 3/2✓, 2✓ but NOT greater than 3.</p><p><strong>Answer: (A),(B),(C)</strong></p><div class="trap-box"><strong>Trap:</strong> Stop at the half-step relation. Full period visible only after converting to unit-step recurrence.</div><div class="key-concept"><strong>Key Concept:</strong> Half-step functional equation → unit recurrence → period 3</div></div>
Correct Answer: B