# Calculating the maxima of a function subject to a constraint

The first figure below shows the plots of a function $$f(x)$$ plotted for different values of a parameter $$\lambda$$, i.e., $$f(x)=f(x,\lambda=\lambda_i);i=1,2,3,..;\lambda$$ is maximum for the topmost curve and decreases downwards. The second figure shows the values of $$\lambda$$ that corresponds to the extrema.

Each curve has one maxima and one minima. The dotted curve corresponds to the case that has no maxima or minima, i.e., the value of $$\lambda$$ corresponding to the dotted curve is the threshold for existence of any extrema.

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 which matches exactly with that of particle dynamics. In this sense, figure-2 is the same for particle dynamics as well as hydrodynamics.

The following are my goals:

• To find the threshold value of $$\lambda$$ (corresponding to the black dotted curve): This I had already calculated by finding the minima 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 on the $$\lambda-x$$ curve corresponds to extremum, I do not have any idea about how to use the constraint $$f(x)=1$$ as the expression for $$f(x)$$ is not available.

• In figure-1, all the curves saturates at $$f(x)=1$$ at large values of $$x$$. So the constraint $$f(x)=1$$ is satisfied for the black(solid) curve as seen in the figure at $$x=2.91$$ and also for all the curves at large $$x$$. So I think that the asymptotic behaviour of $$f(x)$$ might be related to the values of $$\lambda$$.
• In figure-2, initially $$\lambda$$ decreases steeply with $$x$$ and after the minima, $$\lambda$$ varies as $$\lambda=\sqrt x$$.