# Algorithms: calculation of the maximums of a function subject to a restriction

The first figure below shows the graphs of a function $$f (x)$$ plotted for different values ​​of a parameter $$lambda$$that is to say $$f (x) = f (x, lambda = lambda_i); i = 1,2,3, ..; lambda$$ It is maximum for the upper curve and decreases down. The second figure shows the values ​​of $$lambda$$ That corresponds to the extremes.

Each curve has a maximum and a minimum. The dotted curve corresponds to the case that has no maximum or minimum, that is, the value of $$lambda$$ corresponding to the dotted curve is the threshold for the existence of any end.

Note that Figure-1 is obtained from particle dynamics where we have an expression for $$f (x)$$ and we can obtain figure 2 by calculating the extreme values. However, in hydrodynamics we can only obtain Figure 2 that exactly matches that of particle dynamics. In this sense, Figure 2 is the same for particle dynamics and hydrodynamics.

The following are my goals:

• To find the threshold value of $$lambda$$ (corresponding to the black dotted curve): This had already been calculated by finding the minimums of the $$lambda-x$$ curve.
• To find the value of $$lambda$$ for which $$f (x) = 1$$ (corresponding to the black solid curve): Since all the points in the $$lambda-x$$ the curve corresponds to the end, I have no idea how to use the constraint $$f (x) = 1$$ as the expression for $$f (x)$$ not available.

• In Figure 1, all curves are saturated in $$f (x) = 1$$ at large values ​​of
$$x$$. Then the restriction $$f (x) = 1$$ it is satisfied for the black (solid) curve as seen in the figure in $$x = 2.91$$ and also for all curves in general $$x$$. So I think the asymptotic behavior of $$f (x)$$ could be related to the values ​​of $$lambda$$.
• In figure 2, initially $$lambda$$ abruptly decreases with $$x$$ and after the minimums, $$lambda$$ Varies according $$lambda = sqrt x$$.