MonotonicityHard
Question
If f (x) = a{aIxI sin x} ; g(x) = a[aixi sin x] for a > 0,a ≠ 1and x ∈ R, where { } & [ ] denote the fractional part and integral part functions respectively, then which of the following statements holds good for the function h (x), where (ln a)h(x) = (ln f (x) + ln g(x)) -
Options
A.′h′ is even and increasing for a > 1
B.′h′ is odd and decreasing for a <1
C.′h′ is even and decreasing for a < 1
D.′h′ is odd and increasing fro a > 1
Solution
(ln a) h(x) = lna{a|x|sgn x} + ln a[a|x|sgn x]
(ln a) h(x) = ({a|x|sgn x} + [a|x|sgn x]) ln a
(ln a) h(x) = (|x|sgn x) (ln a)
⇒ h(x) = a|x|sgn (x)
If a > 1 ⇒ ′h′ is odd & increasing 0 < a < 1 ⇒ ′h′ is odd but neither increasing nor decreasing.
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