If the inequality \(\sin ^{2}x+a\cos x+a^{2}\geq1+\cos x\) holds for any \(x\in \textbf{R}\),then find the range of values for negative \(\textbf{a}\).
Given \(A\equiv \left ( 0,2 \right )\) and two points \(\textbf{B}\) and \(\textbf{C}\)on the parabola \(y^{2}=x+4\) such that \(AB\perp BC\), determine the range for the y-coordinate of point \(\textbf{C}\).
If \(log_4\left ( x+2y \right )+log_4\left ( x-2y \right )=1\), then the minimum value of \(\left | x \right |-\left | y \right |=\).
