Application of DerivativeHard
Question
If f(x) = a{a|x|sin x}; g(x) = a[a|x|sin x] for a > 1, 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
B.′h′ is odd and decreasing
C.′h′ is even and decreasing
D.′h′ is odd and increasing
Solution
f(x) = a{a|x|sgn x}, g(x) = a[|x| sgn x]
for a > 1, a ≠ 1 and x ∈ R
ln a h(x) = ln f(x) + lng(x)
⇒ (ln a) h(x) ={a|x| sgnx} + lna + [a|x|sgnx]
⇒h(x) = a|x| sgn x
Now h(-x) = a|-x| sgn (-x) = -h(x)
h(x) is an odd function
Also graph of h(x) is
It is clear from the graph that h(x) is an increasing function
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