What does ∇ f=λ∇ g mean geometrically at an extremum of f on the curve g=k?

The level curve of f is tangent to the curve g=k

The curve g=k has a critical point of f there

f is constant on the whole plane

Both gradients are perpendicular

Exercise · ZW-LAGRANGE-MULTIPLIERS-009 · ★★★ Advanced

What does \(\nabla f=\lambda\nabla g\) mean geometrically at an extremum of \(f\) on the curve \(g=k\)?

Choose one option.

Show answer

C. The level curve of \(f\) is tangent to the curve \(g=k\)

Explanation

The gradients are parallel, and each gradient is perpendicular to its level curve; hence the level curves of \(f\) and \(g\) share a tangent at that point.

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Review the concept: Lagrange multipliers

The method of Lagrange multipliers finds the extrema of a function \(f\) on the curve \(g=k\) among the points where the gradient of \(f\) is a multiple \(\lambda\) of the gradient of \(g\).

\[\nabla f=\lambda\,\nabla g,\qquad g=k\]

What does \(\nabla f=\lambda\nabla g\) mean geometrically at an extremum of \(f\) on the curve \(g=k\)?

Explanation

Review the concept: Lagrange multipliers

More practice problems: lagrange multipliers