5. Given a pair of real numbers $(x, y)$ such that $a=2 x^{2}-y^{2}+\frac{3}{2}$ and $b=y^{4}+4 x+2$ have a maximum value not exceeding 1, then the range of $(x, y)$ that satisfies the condition is $\qquad$ .
Answer: $\left\{\left(-\frac{1}{2}, 1\right),\left(-\frac{1}{2},-1\right)\right\}$
Difficulty: Medium
Examining: Inequality
Analysis: From the given, $\left\{\begin{array}{l}a=2 x^{2}-y^{2}+\frac{3}{2} \leq 1 \\ b=y^{4}+4 x+2 \leq 1\end{array}\right.$
$$
\begin{array}{l}
\Rightarrow 2 a+b=4 x^{2}+4 x+1+y^{4}-2 y^{2}+1 \leq 0 \\
\Rightarrow(2 x+1)^{2}+\left(y^{2}-1\right) \leq 0 \\
\Rightarrow x=-\frac{1}{2}, y= \pm 1
\end{array}
$$

Conversely, $x=-\frac{1}{2}, y= \pm 1 \Rightarrow a=b=1$, satisfying the conditions