Optimization problem

Question

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

Answer 1

Here is a sketch of the function:

The distance is found using the Pythagorean Theorem where the two sides of the triangle are "x" and "y" meaning d = √(x² + y²) however, y = √(1 - x) so replacing "y" with "√(1 - x)" we get:

d = √(x² + (√(1 - x))²)

d = √(x² + (1 - x))

d = √(x² - x + 1)

d = (x² - x + 1)^1/2

To minimize the distance, take the derivative (using the power rule and the chain rule) and set it equal to zero

d' = 1/2(x² - x + 1)^-1/2(2x - 1)

0 = 1/2(x² - x + 1)^-1/2(2x - 1)

Since "x² - x + 1" is in the denominator (negative exponent), then it cannot contribute to the product being equal to zero, so we can just ignore it

0 = 2x - 1

2x = 1

x = 1/2

The point is (x, y) so we must plug in 1/2 into the original function √(1 - x) to get the y-value:

√(1 - 1/2) = √1/2 = √1/√2 = 1/√2 = √2/2 = 0.707

So the closest point to the origin is (0.5, 0.707)