Here’s how you can achieve both parts (i) and (ii) in Mathematica for the function $g(x, y) = 3x \cos(x + y^2)$.

Part (i): Creating a Contour Plot

To create a contour plot with the specified range and number of contours, and to add a legend, use the following code in Mathematica:

mathematicaCopy code
(* Define the function *)
g[x_, y_] := 3 * x * Cos[x + y^2]

(* Create the contour plot with 16 contours and a legend *)
ContourPlot[g[x, y], {x, -4.5, 4.5}, {y, -2.5, 2.5}, 
 Contours -> 16, ColorFunction -> "Rainbow", 
 PlotLegends -> Automatic, 
 PlotRange -> All, FrameLabel -> {"x", "y"}]

• Contours -> 16 specifies the number of contours to show.
• `Color