Let `f:R->R` be a twice differential function such that `f(pi/4)=0,f((5pi)/4)=0 and f(3)=4`, then show that there exist a `cin(0,2pi)` such that `f^(primeprime)(c)+sinc-cosc<0`

Explanation

Consider g(x) = f(x) =sinx +cosx
`rArr " " "g(x) =f(x) -cosx -sinx"`
`rArr" ""g(x) =f(x) + sin x -cosx"`
By LMVT
`(g(3)-g((pi)/(4)))/(4) =g(c), (pi)/(4) lt c , lt 3 " " and "(g((5pi)/(4))-g(3))/((5pi)/(4)-3)=g(c_(2)), 3ltc_(2)lt.(pi)/(4)`
`g(c_(1)) gt 0 g(c_(2)) lt0`
By LMVT
`(g'(c_(2)) -g'(c_(1)))/(c_(2)-c_(1))=g''(c), " " c_(1)lt clt c_(2) " "rArrg''(c)lt0 " "rArrf''(c) + sinc - cosc lt 0`
for some `c in (c_(1) ,C_(2)) c in (0,2pi)`