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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 : sgn alpha denote signum function of alpha.] Number of points where G(x)=|f(x)|+f(|x|) is non-differentiable in (-3pi,3pi), is

Answer»

6
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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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