Optimization problem 2.?

The equation for the distance between two points is:

d = √[(x₁ - x₂)² + (y₁ - y₂)²]

We want to find the minimum distance between the origin (0, 0) and the point (x, 3x - 2)

If we set up the origin as point two in the equation above and substitute the variables and expressions into the distance equation we get:

d = √[(x - 0)² + (3x - 2 - 0)²]

simplify:

d = √[x² + (3x - 2)²]

d = √(x² + 9x² - 12x + 4)

d = √(10x² - 12x + 4)

The "x" that makes "d" a minimum can be found by solving for the zero of the first derivatiave.  We'll need the chain rule:

d = √u and u = 10x² - 12x + 4

dd/du = 1/(2√u) and du/dx = 20x - 12

dd/dx = dd/du * du/dx

dd/dx = 1/(2√u) * (20x - 12)

dd/dx = (20x - 12) / (2√u)

dd/dx = 2(10x - 6) / (2√u)

dd/dx = (10x - 6) / √u

dd/dx = (10x - 6) / √(10x² - 12x + 4)

0 = (10x - 6) / √(10x² - 12x + 4)

Multiply both sides by the denominator and solve for x:

0 = 10x - 6

6 = 10x

3/5 = x

I would keep answers as fractions unless told otherwise, but yes, 0.6 is the same as 3/5 and we get the same answer.

We can solve for the "y" value to find the exact point that is closest to the origin:

f(x) = 3x - 2

f(3/5) = 3(3/5) - 2

f(3/5) = 9/5 - 2

f(3/5) = 9/5 - 10/5

f(3/5) = -1/5

The point closest to the origin is (3/5, -1/5)

yes it is............