Given f ( x ) = array ll x ^ 2 e ^ 2 ( x - 1 ) , 0 ≤ x ≤ 1 . asgn (...

$f$\left({1}^{-}\right) = 1 = f\left(1\right)$and$f\left({1}^{+}\right) = a + b$<br /><br />For continuity at$x = 1, a + b = 1 \ldots \ldots \left(i\right)$<br /><br />In$0 < x < 1, {f}^{'}\left(x\right) = 2x{e}^{2\left(x - 1\right)} + 2{x}^{2}{e}^{2\left(x - 1\right)}$<br /><br />$∴ {f}^{'}\left({1}^{-}\right) = 4$<br /><br />In$1 < x < 2, {f}^{'}\left(x\right) = - 2asin \left(2x - 2\right) + 2bx$<br /><br />$∴ {f}^{'}\left({1}^{+}\right) = 2b$<br /><br />For differentiability at$x = 1, 2b = 4$. This with$( i )$gives the values$a = - 1, b = 2$<br /><br />then$| b - a | = 3