Let `f(x) = (1 - x)^2 sin^2 x + x^2 ` for all x ∈ R, and let `g(x) = ∫((2(t - 1))/(t + 1) - ln t)f(t)dt` for t ∈ [1, x] for all x ∈ (1, ∞).Which of the following is true ?

Options

A. f is continuous at `x = pi//2`
B. f has an irremovable discontinuity at `x=pi//2`
C. f has a removable discontinuity at `x=pi//2`
D. none of these

Answer

B

Explanation

`f(x)={{:((2cosx-sin2x)/((pi-2x)^(2))",",x le(pi)/(2)),((e^(-cos)-1)/(8x-4pi)",",x gt(pi)/(2)):}`
L.H.L. at `x=pi//2`
`underset(hrarr0)(lim)(2sinh-sin2h)/(4h^(2))=underset(hrarr0)(lim)(2sinh(1-cosh))/(4h^(2))=0`
`"R.H.L."=underset(hrarr0)(lim)(e^(sinh)-1)/(8((pi//2)+h)-4pi)`
`=underset(hrarr0)(lim)(e^(sin h-1))/(8h).(sin h)/(sin h)=(1)/(8)`
`rArr" h(x) has irremovable discontinuity at x "=pi//2`.