Find a piecewise smooth parametrization of the path C?

Note that all of the suggested options are linear in #t#, so it suffices to check the endpoints in each interval.

First option

Putting #t=0# and #t=5# into the first expression, we get:

#(5-(color(blue)(0)))hat(i)+(4-4/5(color(blue)(0)))hat(j) = (5, 4)#

#(5-(color(blue)(5)))hat(i)+(4-4/5(color(blue)(5)))hat(j) = (0, 0)#

So this traverses the segment between #(0, 0)# and #(5, 4)# backwards. So we can reject the first option.

Second option

Putting #t=0# into the first expression, we get:

#5hat(i)+(9-(color(blue)(0)))hat(j) = (5, 9)#

This point is not on the curve, so we can reject the second option.

Third option

The first expression is fine, but putting #t=5# into the second expression gives us the point #(9, 0)# which is not on the curve.

Fourth option

This one works:

• Evaluating the endpoints of the first expression gives #(0, 0)# and #(5, 4)#

• Evaluating the endpoints of the second expression gives #(5, 4)# and #(5, 0)#

• Evaluating the endpoints of the third expression gives #(5, 0)# and #(0, 0)#

So this parametrisation traverses the curve as required.