If a function \(f(x)\) defined by \( f(x)= \begin{cases}\mathrm{ae}^x+\mathrm{be}^{-x}, & -1 \leq x<1 \\ \mathrm{…

If a function \(f(x)\) defined by
\(
f(x)= \begin{cases}\mathrm{ae}^x+\mathrm{be}^{-x}, & -1 \leq x<1 \\ \mathrm{cx}^2 & , 1 \leq x \leq 3 \\ \mathrm{a} x^2+2 \mathrm{cx}, & 3<x \leq 4\end{cases}
\)
be continuous for some \(a, b, c \in \mathbf{R}\) and \(f^{\prime}(0)+f^{\prime}(2)=\mathrm{e}\), then the value of \(\mathrm{a}\) is :

[JEE Main 2020, 2 Sep (Shift 1)]