A road profile in a design software is represented, for large values of x, by the polynomial r(x)=0.5x6+c1x4+c2x2+c3, where c1,c2,c3∈R. Without calculating the constants, answer the following:
a. (i) State whether the graph of r(x) is symmetric about the y-axis, symmetric about the origin, or neither. Give a brief justification.
b. (ii) Determine the direction in which the road profile rises or falls as one moves far to the east (x→+∞) and far to the west (x→−∞).
c. (iii) If the leading coefficient 0.5 is replaced by −0.5, describe precisely how the answers in part (ii) will change.