Find the points on the curve x^3 - y^2 = 1 which are closest to the point (2, 5)?

Answer 1

Looking at the graph of y = SQRT(x³-1), there appears to be one point that is closest.

the distance between the curve and the point (2,5) is d = SQRT( (x-2)² + (y-5)² )

d = SQRT( (x-2)² + (SQRT(x³-1)-5)² ) = SQRT( x³ + x² -4x + 28 -10(SQRT(x³-1))

You can also use d² to find the minimum value. So

d² = x³ + x² - 4x + 28 - 10(SQRT(x³-1)) The take the derivative of d² & set to zero

d(d²)/dx = 3x² + 2x -4 - 15x²/SQRT(x³-1) = 0

Solving for d(d²)/dx = 0, you find that x = 2.838 , substituting into the equation and solving for y

y = 4.676