Find the points on the curve xy^2 = 54 that are nearest to the origin using Lagrange multipliers.

Understand the Problem

The question asks to find the points on the curve xy^2 = 54 that are nearest to the origin using Lagrange multipliers. This is an optimization problem where we need to minimize the distance from the origin (0,0) to a point (x, y) on the curve, subject to the constraint xy^2 = 54.

Answer

$(3, 3\sqrt{2})$ and $(3, -3\sqrt{2})$

Steps to Solve

Define the objective function

We want to minimize the distance from the origin to a point $(x, y)$. The distance formula is $\sqrt{x^2 + y^2}$, but to simplify calculations and avoid square roots, we can minimize the square of the distance, which is $f(x,y) = x^2 + y^2$.

Define the constraint function

The constraint is given by the equation $xy^2 = 54$. We can rewrite this as $g(x,y) = xy^2 - 54 = 0$.

Set up the Lagrange multiplier equations

We need to solve the following system of equations:

$\nabla f(x,y) = \lambda \nabla g(x,y)$
$g(x,y) = 0$

First, find the gradients:

$\nabla f(x,y) = \langle 2x, 2y \rangle$
$\nabla g(x,y) = \langle y^2, 2xy \rangle$

Now, set up the equations:

$2x = \lambda y^2$ (1)
$2y = \lambda (2xy)$ (2)
$xy^2 = 54$ (3)

Solve for $\lambda$ from equation (2)

From equation (2), $2y = 2\lambda xy$. If $y \neq 0$, we can divide both sides by $2y$ to get $1 = \lambda x$, so $\lambda = \frac{1}{x}$. Note that $y=0$ is not possible since from (3), $xy^2 = 54$, implies that neither $x$ nor $y$ can be zero.

Substitute $\lambda$ into equation (1)

Substitute $\lambda = \frac{1}{x}$ into equation (1): $2x = \frac{1}{x} y^2$, which simplifies to $2x^2 = y^2$.

Substitute $y^2$ into equation (3)

Substitute $y^2 = 2x^2$ into equation (3): $x(2x^2) = 54$, which simplifies to $2x^3 = 54$.

Solve for $x$

Divide both sides by 2: $x^3 = 27$. Taking the cube root, we get $x = 3$.

Solve for $y$

Substitute $x = 3$ into $y^2 = 2x^2$: $y^2 = 2(3^2) = 2(9) = 18$. Therefore, $y = \pm \sqrt{18} = \pm 3\sqrt{2}$.

Find the points

The points on the curve closest to the origin are $(3, 3\sqrt{2})$ and $(3, -3\sqrt{2})$.

The points on the curve $xy^2 = 54$ that are nearest to the origin are $(3, 3\sqrt{2})$ and $(3, -3\sqrt{2})$.

More Information

The method of Lagrange multipliers is a powerful technique for solving constrained optimization problems. It allows us to find the maximum or minimum of a function subject to one or more constraints. By introducing Lagrange multipliers, we can transform the constrained optimization problem into an unconstrained problem, which can then be solved using standard calculus techniques.

Tips

A common mistake is forgetting to consider both positive and negative roots when solving for $y$. In this problem, $y^2 = 18$, so it is crucial to consider both $y = 3\sqrt{2}$ and $y = -3\sqrt{2}$. Another common mistake is to incorrectly compute the gradients or set up the Lagrange multiplier equations, leading to incorrect solutions. Also, it can be easy to make algebraic errors.

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