In the first three chapters, we established the most useful necessary conditions for a weak minimum of the basic problem in the calculus of variations and its variants. Among these conditions, the Euler–Lagrange equation enables us to narrow our search for a local minimum point to a small group of PWS functions called extremals. However, the fact that y ^ ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749821/5a9bd1b0-aa97-4dfe-adb3-8f69a65a6308/content/eq648.tif"/> satisfies the Euler–Lagrange equation does not necessarily make it the solution of our problem, just as the condition f′(x ₀) = 0 does not necessarily imply that f(x ₀) is a local minimum for f(x). For the problem of extremizing an ordinary function f(x), we must examine f″(x ₀) to decide whether x ₀ is a minimum point, a maximum point, or a point of inflection of f(x). Analogously, we need to examine the second variation to learn whether an extremal y ^ ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749821/5a9bd1b0-aa97-4dfe-adb3-8f69a65a6308/content/eq649.tif"/> is a minimum point of our variational problem.*