Critical point Definition and 53 Threads

I have few questions about extrema of fonctions of two variables. It is written in my textbook: "At a local maximum, the gradient vector must be nul or undefined. A similar reasoning shows that the gradient must be nul at a local minimum." Actually there was no preceeding reasoning to this...

let f(x)=sin(1/x)*x^2 for x not 0, and f(0)=0. show that x=0 is a critical point for f which is neither a local minimum, a local maximum, nor an inflection point. well I tried differentiating this, and got f'=-cos(1/x) +2xsin(1/x). to find a critical point i make f'=0. Not sure how to do...

phase equilibrium -- beyond critical point In a phase diagram, at critical point, liquid phase is indistinguishable from vapour, and beyond which, only vapour can be found. Matter has four states, liquid, solid, gas and plasma. Can we include plasma state in a phase diagram? If yes, how should...