1.

Consider f(x)=Lim_(nto oo)((a^(n)+b^(n))^((1)/(n))sinx+{e^(x)}^(n))([(1)/(ncot^(-1)n)]+1),AAx inR where agtbgt0. [Note : {k} and [k] denotes fractinal part of k and greatest interger less than or equal to k respectively.] If H(x)=sgn (f(x)-3) has exactly one point of discontinuity AA x in[0,2pi], then number of integral value of a, is

Answer»

1
2
0
infinite

Solution :`F(x)=underset(ntooo)Lim((a(1+((b)/(a))^(n))^((1)/(n)))sinx+{e^(x)}^(n))([((1)/(n))/("tan"^(-1)(1)/(n))]+1)`
`:.""f(x)=(asinx+0)(1+1)`
`:.""f(x)=2asinx`
(i) `H(x)=SGN (2asinx-3)` has exactly one POINT of discontinuty in `[0,2pi]`, then `2A sinx-3=0` must have one real root in `[0,2pi],sin(3)/(2a)`
`:." "a=(3)/(2)` only
`:.""` Number integral value of a is ZERO.
`G(x)=|2asinx|+2asin|x|`
(ii)
Number of non-differential point of G(x) is `x=-2pi,-pi,0,pi,2pi`
separately we can prove that G(x) is non-differentiable at x=0.


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