4d.
\(P=\dfrac{4\sqrt{x}+1}{2\sqrt{x}+3}\)
\(=\dfrac{2\left(2\sqrt{x}+3\right)}{2\sqrt{x}+3}-\dfrac{5}{2\sqrt{x}+3}\)
\(=2-\dfrac{5}{2\sqrt{x}+3}\)
\(\ge2-\dfrac{5}{2.0+3}=\dfrac{1}{3}\)
\(\Rightarrow minP=\dfrac{1}{3}\Leftrightarrow x=0\)
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