Optimization problem

Question

which point squareroot (1-x) is closest to the origin. Rund your answer to three decimal places

Answer 1

The distance d from the origin to (1-x)^1/2 is sqrt(x² -x +1).

Differentiate d with respect to x and set the derivative equal to zero.

You will find the value of x at which the distance is a minimum is 1/2.

Answer 2

y = √(1-x)

distance to origin:

d(x) = √((x-0)²+(y-0)²) = √(x²+y²) = √(x²+1-x)

To minimize, find d'(x) and set it to zero then solve for x

d'(x) = (1/(2√(x²-x+1))(2x-1)

2x-1 = 0

x = 1/2

y=√(1-1/2) = 1/√2 = √2/2

Point = (1/2, √2/2)

Here is the graph

https://www.desmos.com/calculator/fonygory1r