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(Solved): B4 For a nonnegative integer n and a strictly increasing sequence of real numbers t_(0),t_(1)dots,t_ ...
B4 For a nonnegative integer
nand a strictly increasing sequence of real numbers
t_(0),t_(1)dots,t_(n), let
f(t)be the corresponding real-valued function defined for
1>=10by the following properties: (a)
f(t)is continuous for
t>=t_(0), and is twice differentiable for all
t>i_(0)other than
t_(1),dots,t_(n); (b)
f(t_(0))=(1)/(2); (c)
\lim_(t->t_(k)^(+))f^(')(t)=0for
0<=k<=n; (d) For
0<=k<=n-1, we have
f^('')(t)=k+1when
f^('')(t)=n+1t>t_(n)nt_(0),t_(1)dots,t_(n)t_(k)>=t_(k-1)+11<=k<=nTf(t_(0)+T)=2023t_(k), and f^('')(t)=n+1 when t>t_(n).
Considering all choices of n and t_(0),t_(1)dots,t_(n) such that t_(k)>=t_(k-1)+1 for 1<=k<=n, what is the least possible value of T for which f(t_(0)+T)=2023 ?