Existence or properties of extrema via abstract/theoretical argument
The question requires proving existence or uniqueness of a minimizer/maximizer using theoretical tools (compactness, convexity, Lagrange multipliers) rather than explicit computation.
A polynomial $P(x)$ with real coefficients and of degree four satisfies the inequality $$P(x) \geq x$$ for every real number $x$. $$\begin{aligned}
& P(1) = 1 \\
& P(2) = 4 \\
& P(3) = 3
\end{aligned}$$ according to, $\mathbf{P(4)}$ is equal to what?