Question

2. Find the point on the line 6x+y=9 that is closest to the point (-3,1).

a. Find the objective function.

b. Find the constraint.

c. Find the minimum (You need to specify your method)

Answer 1

Let the point be P(x, y).

As P lies on the line 6x + y = 9,

So, the coordinates of the point P is (x, 9-6x).

The distance between the points P(x, 9-6x) and (-3, 1) is

Note that d is minimum whenever is minimum.

Thus, the objective function f(x) is

There is no constraint on x. x can be any real number lying on the interval (-, ).

To minimize the objective function f(x), first we have to find the critical point(s) and then use the second derivative test.

f(x) is differentiable for all real numbers. Thus, the critical point(s) can only be found by setting f'(x) to 0.

By the second derivative test,

As there is only one local minimum point, the point is also the global minimum.

At x = 45/37,

Hence, the point on the line 6x + y = 9 that is closest to the point (-3, 1) is (45/37, 63/37).