If `f(x)={(((1-sin^(3)x))/(3cos^(2)x)",",x lt (pi)/(2)),(a",",x=(pi)/(2)),((b(1-sinx))/((pi-2x)^(2))",",x gt (pi)/(2)):}`
is continuous at `x=(pi)/(2)`, then the value of `((b)/(a))^(5//3)` is

Answer

32

Explanation

`LHL = RHL = f(pi/2) rArr lim_(x rarr (pi)/(2))f(x) = f(pi/2)=lim_(x rarr (pi^+)/(2)) f(x)`
`rArr lim_(h rarr o)f(pi/2 -h)=a=lim_(h rarr 0)f(pi/2 +h) rArr`
`lim_(h rarr 0) ((1-cos^3h))/(3 sin ^2h) = a= lim_(h rarr 0)(b(1-cosh))/(4h^2)`
`rArr lim_(h rarr 0)((1-cos h)(1+cos h +cos^2h))/(3(1-cos h)(1+cos h))=a=lim_(h rarr 0) (b(1-cos h))/(4h^2)`
`rArr (3)/(3(2)) = a = (b)/(8) or (b)/(a) = 8 therefore `
`(b/a)^(5//3) = (8)^(5//3) =2^5 = 32`