Now
| δu = Σ | δu | δx1 + 1⁄2 { Σ | δ2u | δx12 + 2Σ | δ2u | δx1δx2 ... } + .... |
| δx1 | δx12 | δx1δx2 |
The sign of this expression in general is that of Σ(δu/δx1)δx1, which cannot be one-signed when x1, x2, ... xn can take all possible values, for a set of increments δx1, δx2 ... δxn, will give an opposite sign to the set −δx1, −δx2, ... −δxn. Hence Σ(δu/δx1)δx1 must vanish for all sets of increments δx1, ... δxn, and since these are independent, we must have δu/δx1 = 0, δu/δx2 = 0, ... δu/δxn = 0. A value of u given by a set of solutions of these equations is called a “critical value” of u. The value of δu now becomes
| 1⁄2 { Σ | δ2u | δx12 + 2 Σ | δ2u | δx1δx2 + ... }; |
| δx12 | δx1δx2 |
for u to be a maximum or minimum this must have always the same sign. For the case of a single variable x, corresponding to a value of x given by the equation du/dx = 0, u is a maximum or minimum as d2u/dx2 is negative or positive. If d2u/dx2 vanishes, then there is no maximum or minimum unless d2u/dx2 vanishes, and there is a maximum or minimum according as d4u/dx4 is negative or positive. Generally, if the first differential coefficient which does not vanish is even, there is a maximum or minimum according as this is negative or positive. If it is odd, there is no maximum or minimum.
In the case of several variables, the quadratic
| Σ | δ2u | δx12 + 2 Σ | δ2u | δx1δx2 + ... |
| δx12 | δx1δx2 |
must be one-signed. The condition for this is that the series of discriminants
| a11 , | a11 a12 | , | a11 a12 a13 | , ... |
| a21 a22 | a21 a22 a23 | |||
| a31 a32 a33 |
where apq denotes δ2u/δapδaq should be all positive, if the quadratic is always positive, and alternately negative and positive, if the quadratic is always negative. If the first condition is satisfied the critical value is a minimum, if the second it is a maximum. For the case of two variables the conditions are