Here I want to prove the uniqueness of the weak solution for the homogeneous Dirichlet boundary value problem:
\begin{equation*}
\left\{
\begin{array}{rl}3u′′ − 2u′ + 3u = f \\
u(0) = u(1) = 0\\
\end{array}\right.
\end{equation*}
for any given $f ∈ C(\bar{I})$ on $I = (0, 1)$ by using Lax-Milgram lemma. So first I have to show the map which is defined by $\alpha : H^1(I) × H^1(I) → \mathbb{R},$
$$\alpha(f, g) = \int_I(3f′g′ − f′g − fg′ + 3fg)$$
is bounded, symmetric and coercive bilinear form. But I couldn't go further the coercive part. Could somebody please give me a hint?