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
Create a free account to view solution
View Solution FreeMore Application of Derivative Questions
If the point (1, 3) serves as the point of inflection of the curve y = ax3 + bx2 then the value of ′a′ and &...The radius of a right circular cylinder of greatest curved surface which can be inscribed in a given right circular cone...Let h(x) = f(x) - (f(x))2 + (f(x))3 for every real number x. Then...If a, b > 0 and ᐃ (x) = , then -...If the tangent at P of the curve y2 = x3 intersects the curve again at Q and the straight lines OP, OQ make angles α...