Final answer:
The non-zero value of Lagrange multiplier λ by itself does not determine the nature of the critical point as minimum, maximum, or saddle point. Additional analysis such as the second derivative test is required to ascertain stability or extremum. In equilibrium analysis, the nature of forces and potential energy behavior are key indicators of stability.
Step-by-step explanation:
The concept of Lagrange multipliers is used in mathematics to find the local maxima and minima of a function subject to equality constraints. The question asks what can be inferred if it is certain that lambda (usually denoted as λ), which is the Lagrange multiplier, is not equal to 0.
However, the value of λ being non-zero does not by itself determine whether a critical point is a minimum, maximum, or saddle point. The function of interest should instead be analyzed by examining the second-order derivatives (i.e. the second derivative test) or further constraints to determine the nature of the critical point.
For the context of potential energy and forces discussed along with equilibrium points, the nature of stability can be assessed. If the force is a restoring force, it implies that the potential energy (U) has a relative minimum at that point, suggesting a stable equilibrium. Conversely, if the force direction is such that it moves the system away from the equilibrium point, the potential energy has a relative maximum there, indicating an unstable equilibrium. Therefore, without additional information, the statement about λ not being zero does not directly determine the nature of the critical point in terms of stability or extremum.