Page 111 - How To Solve Word Problems In Calculus

It is clear, even without precise calculation, that since the crit-
√
3

ical value is positive, S will be positive at 10/ π. Therefore,
this solution yields a relative minimum surface area. Being
the only relative extremum on the interval (0, ∞), S(r) has an
absolute minimum at this location.

EXAMPLE 13
Find the point on the line 3x + y = 6 closest to (2, 3).

Solution
Step1
Let (x, y) be a point on the line.

(x, y)
d
(2, 3)

Step2
The distance between (x, y) and (2, 3) is

2
d = (x − 2) + (y − 3) 2 The distance between
(x 1 , y 1 ) and (x 2 , y 2 )is

2
2
2
= x + y − 4x − 6y + 13 d = (x 2 − x 1 ) + (y 2 − y 1 ) 2
Step3
Since the point (x, y) lies on the line, y = 6 − 3x,

2
2
d = x + (6 − 3x) − 4x − 6(6 − 3x) + 13

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