Let $(u_k)_{k\geq 0}$ and $(v_k)_{k\geq 0}$ be two sequences with $u_0 \lt v_0$ and $u_1\gt 0,v_1\gt 0$.
My question : is there an $f\in{\mathcal C}^{\infty}([0,1],{\mathbb R})$ with $f^{(k)}(0)=u_k$ and $f^{(k)}(1)=v_k$, for every $k\geq 0$, and $f$ is increasing on $[0,1]$ ?
My thoughts : if ones removes the condition that $f$ be increasing on $[0,1]$, it seems that the Whitney extension theorem should be useful. However the theorem yields a function in ${\mathcal{C}}^m$ for a finite $m$, not in ${\mathcal{C}}^{\infty}$. It is also unclear to me how to construct a monotonic solution from non-monotonic ones.
I am not sure if this is overcomplicating things. Borel's lemma for smooth functions might be used to construct one such function. By the lemma we have a smooth functions $g_0, g_1 \in C^\infty$ such that $g_0^{(k)}(0) = u_{k+1}$ and $g_1^{(k)}(1) = v_{k+1}$ for all $k \ge 0$.
In what follows, I want the profile of $f'$ to be disjoint bump functions around $0$ and $1$ with prescribed derivatives at these points. Then adjust the mass of each of these bumps to get the required vales $u_0$ and $v_0$ of $f$ at $0$ and $1$. Non negativity of $f'$ in $[0,1]$ ensures $f$ is non-decreasing.
$\underline{\text{Step-1}}$ (construction of bump function near $0$):
We may choose a non-negative test function $\phi_0 \in C_c^{\infty}$ s.t., $\phi_0 \equiv 1$ in a small nbd of $0$, $\phi_0 \ge 0$ and $\operatorname{supp}(\phi_0) \subset (-1/2, 1/2) \cap \{g_0 > 0\}$ (this is possible since $g_0(0) = u_1 > 0$ and hence positive in a small nbd of $0$) and $\displaystyle \int_{-1/2}^{1/2} |g_0\phi_0| \,dx = m_0 < (v_0 - u_0)$ (i.e. with sufficiently small mass). Suppose, $\displaystyle \int_{-1/2}^{0} g_0\phi_0 \,dx = a$ (possibly $a \neq 0$) and $\displaystyle \int_{0}^{1/2} g_0\phi_0\,dx = b$ (note that $|a|, |b| < m_0$). Then we may consider a standard non-negative test function $\psi_0 \in C_c^{\infty}$ with $\operatorname{supp}(\psi_0) \subset (-\infty,-1/2)$ and $\displaystyle \int_{-\infty}^{\infty} \psi_0 \,dx = 1$ and finally consider $G_0 = (g_0\phi_0 - a\psi_0)$. This $G_0$ is non-negative in $[0,1/2)$ and has the property that $\displaystyle \int_{-\infty}^{0} G_0\,dx = 0$ and $\displaystyle \int_{-\infty}^{1/2} G_0\,dx = b$.
$\underline{\text{Step-2}}$ (construction of bump function near $1$):
Similarly, choose a non-negative test function $\phi_1 \in C_c^{\infty}$ s.t., $\phi_1 \equiv 1$ is a nbd of $1$ with $\operatorname{supp}(\phi_1) \subset (1- \epsilon, 1+\epsilon) \cap \{g_1 > 0\}$ (this is also non empty by the assumption $g_1(1) = v_1 > 0$ and hence positive in a nbd of $1$) with $\displaystyle \int_{1-\epsilon}^{1+\epsilon} |g_1\phi_1| \,dx = m_1 < (v_0 - u_0 - m_0)$ (i.e., with small enough mass). Suppose, $\displaystyle \int_{1-\epsilon}^1 g_1\phi_1\,dx = c$ (note that $|c| < m_1$). That leaves us room to add a standard non-negative test function $\psi_1 \in C_c^{\infty}$ with $\operatorname{supp}(\psi_1) \subset (1/2,1-\epsilon)$ and $\displaystyle \int_{1/2}^{1 - \epsilon} \psi_1\,dx = 1$. Finally consider $G_1 = g_1\phi_1 + (v_0 - u_0 - b - c)\psi_1$ (note that $v_0 - u_0 - b - c > 0$ by our assumptions).
This $G_1$ is non-negative in $(1/2,1]$ and has the property that $\displaystyle \int_{1/2}^{1} G_1\,dx = v_0 - u_0 - b$.
Finally, combining the two we have the function $\displaystyle f(x) = u_0 + \int_{-\infty}^{x} (G_0 + G_1)\,dt$ satisfying the required properties.