If `f(2+x)+f(x)=2` for all `xepsilon [0,2]` and `k=int_(0)^(4)f(x)dx-4` such that both roots of `ax^(2)-bx+c=0` lies between `k` & `q` where `q=(alpha)/(beta)` such that `alpha+beta=1` & `alpha^(2)+beta^(2)=1/2` then least positive integral value of `a`_________.

Answer

5

Explanation

`:' k=int_(0)^(2) f(x) dx+int_(2)^(4)f(x)dx-4`
`=int_(0)^(2) f(x)dx+int_(0)^(2) f(x+2)dx-4`
`= int_(0)^(2) 2.dx-4=0`
& `alpha=beta=1/1 impliesq=1`
`:.` roots of `g(x)=ax^(2)-bx+c` lying in `(0,1)`
`impliesg(0).g(1)=C(a-b+c)epsilonN` so least value is "1"
`=a^(2)alpha_(1)beta_(1)(1-alpha_(1))(1-beta_(1))` as `alpha_(1), beta_(1)` are roots of `ax^(2)-bx+c=0`
`g(0)g(1) lt (a^(2))/16 :' alpha_(1) (1-alpha_(1))ge 1/4`
`implies agt4 ` or `alt -4` & `alpha_(1)!=beta_(1)`
`implies` least value of `a=5`