Let f : \mathbb{R} \to \mathbb{R} be a real valued function such that f(10 + x) = f(10 - x) \forall x \in \mathbb{R} and f(20 + x) = -f(20 - x) \forall x \in \mathbb{R}. Then which of the following statements is true -
f(x) is odd and periodic
f(x) is odd and aperiodic
f(x) is even and periodic
f(x) is even and aperiodic
Step-by-Step Solution
<div class="solution"><p>(I): Let t=cos x∈[-1,1]. t²+t+1∈[3/4,3]. Expression=1+1/(t²+t+1)∈[4/3,7/3] → (Q)</p><p>(II): Simplifies to (cos x-sin x)/√(3(2-u²)) where u=cos x-sin x. Range [0,1/3] → (R)</p><p>(III): x⁶+2x⁴+3x²+1≥1, so expression∈(0,7/3] → (P)</p><p>(IV): (x+1)²+1≥1, so log₈≥0, unbounded → (S)</p><p><strong>Answer: I→Q, II→R, III→P, IV→S</strong></p><div class="trap-box"><strong>Trap:</strong> In (II), domain restrictions on sin x and cos x are critical.</div><div class="key-concept"><strong>Key Concept:</strong> Match-list range problems — reduce each to one-variable range problem</div></div>
Correct Answer: A