interpolation and approximation [closed]

Question

Given a function $f$ in C^k[a,b], can we always construct function $g \neq f$ such that $g(x) \ge f(x)$ for all $x \in [a,b]$, $f^{(m)}(a)=g^{(m)}(a)$ and $f^{(m)}(b)=g^{(m)}(b)$ for $m=0,1,\dots, k$ ? This is not a home work problem, but it is of interesting in a one sided approximation problem. Of course, in the absence of the condition $g \ge f$, Hermite interpolant to $f$ with knots $a$ and $b$ will do.