Problem source

\begin{questionparts} 
\item Find the real values of $x$ for which 
\[
x^{3}-4x^{2}-x+4\geqslant0.
\]
\item Find the three lines in the $(x,y)$ plane on which 
\[
x^{3}-4x^{2}y-xy^{2}+4y^{3}=0.
\]
\item On a sketch shade the regions of the $(x,y)$ plane for which
\[
x^{3}-4x^{2}y-xy^{2}+4y^{3}\geqslant0.
\]
\end{questionparts}

Solution source

\begin{questionparts}
\item $\,$
\begin{align*}
&& 0 & \leq x^3 - 4x^2 - x + 4 \\
&&&= (x-1)(x^2-3x-4) \\
&&&= (x-1)(x-4)(x+1) \\
\Leftrightarrow && x &\in [-1, 1] \cup [4, \infty) 
\end{align*}

\item $\,$ \begin{align*}
&& 0 &= x^{3}-4x^{2}y-xy^{2}+4y^{3} \\
&& 0 &= (x-y)(x-4y)(x+y)
\end{align*}

Therefore the lines are $y = x, 4y = x, y=-x$.

\item 

\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){2*(#1)*((#1)^2 - 5)/((#1)^2-4)};
    \def\xl{-5};
    \def\xu{5};
    \def\yl{-5};
    \def\yu{5};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);

        \draw[thick, blue] (\xl, {\xl}) -- (\xu, {\xu});
        \draw[thick, blue] (\xl, {-\xl}) -- (\xu, {-\xu});
        \draw[thick, blue] (\xl, {0.25*\xl}) -- (\xu, {0.25*\xu});

        \filldraw[blue, opacity = 0.2] (\xl, {\xl}) -- (\xl, {0.25*\xl}) -- (0,0) -- cycle;
        \filldraw[blue, opacity = 0.2] (\xl, {-\xl}) -- (\xu, {\xu}) -- (0,0) -- cycle;
        \filldraw[blue, opacity = 0.2] (\xu, {0.25*\xu}) -- (\xu, {-\xu}) -- (0,0) -- cycle;
        
    \end{scope}
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
    
    \end{tikzpicture}
\end{center}



(quickest way to see this is to check the $x$ or $y$-axis)
\end{questionparts}