Why are the gradients of the objective and constraint functions parallel at the extremum points in the Lagrange multiplier method?

Conditions

• The functions $f$ and $g$ are differentiable.
• The extremum occurs at a point where the constraint curve is smooth (non-zero gradient for $g$).
• The level curves of $f$ are tangent to the constraint curve $g=0$.

Reasoning, step by step

1. Identify the objective function $f(x,y)$ and its level curves.
2. Identify the constraint function $g(x,y)=0$ and its curve.
3. Observe that at the extremum, a level curve of $f$ touches the constraint curve tangentially.
4. Recall that the gradient of a function is perpendicular to its level curves.
5. Conclude that since both gradients are perpendicular to the same tangent line, they must be parallel to each other.
6. Express this parallel relationship mathematically as $\nabla f = \lambda \nabla g$.

Example

The video uses $f(x,y) = x^2 + y^2$ (level curves are circles) and $g(x,y) = x^2 + y^2/2 - 1 = 0$ (constraint is an ellipse). At the points where the circles are tangent to the ellipse, the blue arrows ($\nabla f$) and yellow arrows ($\nabla g$) point in the same or opposite directions, illustrating parallelism.

Common misconceptions

• Believing that the gradients must be equal in magnitude, rather than just parallel.
• Thinking that the gradients are parallel everywhere on the constraint curve, not just at the extremum points.

Watch the explanation

Connected concepts

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• How do you set up the system of equations from the Lagrange multiplier condition?
• What are the specific critical points for the example functions shown in the video?