Let f:[-1,1] → R be defined as f(x)= a x^2+ b x+ c for all x ∈[-1,1], where a, b, c ∈ R such that f(-1)=2, f^′(-1)=1 and for x ∈(-1,1) the maximum…

Let $\displaystyle f:[-1,1] \rightarrow \mathbf{R}$ be defined as $\displaystyle f(x)=\mathrm{a} x^2+\mathrm{b} x+\mathrm{c}$ for all $\displaystyle x \in[-1,1]$, where $\displaystyle \mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbf{R}$ such that $\displaystyle f(-1)=2, f^{\prime}(-1)=1$ and for $\displaystyle x \in(-1,1)$ the maximum value of $\displaystyle f^{\prime \prime}(x)$ is $\displaystyle \frac{1}{2}$. If $\displaystyle f(x) \leq \alpha$, $\displaystyle x \in[-1,1]$, then the least value of $\displaystyle \alpha$ is equal to \_\_\_\_.