A coastal inlet is dredged in such a way that the water depth D (in meters) at a point P along the centerline of the channel depends on the horizontal distance x (in kilometers) from the shoreline according to
D(x)=⎩⎨⎧3x2+1,7.5−2(x−1.5),1+cos(2π(x−3)),0≤x≤1.5,1.5<x<3,3≤x≤4.
Assume the dredged channel ends at x=4km.
a. Part 1: (a) Verify that D is continuous at both x=1.5 and x=3.
b. Part 2: (b) Compute D′(x) on each interval where it exists, and determine all critical points of D in [0,4].
c. Part 3: (c) For each critical point, decide whether it corresponds to a local maximum depth, a local minimum depth, or neither. Support your answer by referencing the sign of D′(x) on the relevant intervals.
d. Part 4: (d) Identify the global maximum depth attained on the entire channel segment 0≤x≤4 and the location(s) where it occurs. Justify your conclusion.
e. Part 5: (e) Interpret your findings from part (d) in practical terms for a cargo vessel planning to navigate the channel.