Given the function
$$f(t) :[0,1] \to \mathbb{R}^{n \times n}, t \mapsto \exp(At) \quad n \in \mathbb{N}, \ A \in \mathbb{R}^{n \times n} $$
and a second-order polynomial $p(t) = A_2t^2 + A_1t + A_0$ such that $f(t) = p(t)$ for $t = 0, \frac{1}{2}, 1$ and $A_2,A_1,A_0 \in \mathbb{R}^{n \times n}$, show that there exists a constant $C>0$ sucht that inequality
$$|f(t) - p(t)| ≤ C \max_{t \in [0,1]}|A^3 \exp(At)|$$
holds. Here we define $|B| = \max_{i,j = 1,2,...,n} |B(i,j)|$ for $B \in \mathbb{R}^{n \times n}$.
My attempt:
I first showed that the polynomial
$$ p(t) = \bigg(2 \exp(A) + 2 I_n - 4 \exp \Big(\frac{1}{2} A \Big) \bigg) t^2 + \bigg( 4 \exp \Big( \frac{1}{2} A \Big) - 3 I_n - \exp(A) \bigg)t + I_n $$
interpolates $f$ at $t = 0, \frac{1}{2}, 1$ and then got
$$|f(t) - p(t)| ≤ \bigg| \exp(At) + 4 \exp \Big(\frac{1}{2} A \Big) t^2 + 3 I_nt + \exp(A)t\bigg | $$
and here I got stuck.