A straight road is represented by the x-axis. A school is located at the point S(−3,0) and a hospital at H(5,0). A bus stop is to be placed at a point P(x,0) on the road so that the sum of walking distances SP+PH is minimised.
a. (i) Express the total walking distance D(x)=SP+PH in terms of x without using piece-wise symbols.
b. (ii) Identify all critical points of D(x), keeping in mind where the derivative may fail to exist.
c. (iii) Determine the value of x that minimises D(x) and state the minimum total distance.