17.2 Minima and maxima, revisited
In a single variable, we have successfully used the derivatives to find the local optima of differentiable functions.
Recall that if f : ℝ →ℝ is differentiable everywhere, then Theorem 87 gives that
(a) f′(a) = 0 and f′′(a)/span>0 implies a local minimum. (b) f′(a) = 0 and f′′(a)/span>0 implies a local maximum.
(A simple f′(a) = 0 is not enough, as the example f(x) = x3 shows at 0.)
Can we do something similar in multiple variables?
Right from the start, there seems to be an issue: the derivative is not a scalar (thus, we can’t equate it to 0).
This is easy to solve: the analogue of the condition f′(a) = 0 is ∇f(a) = (0,0,…,0). For simplicity, the zero vector (0,0,…,0) will also be denoted by 0. Don’t worry, this won’t be confusing; it’s all clear from the context. Introducing a new notation for the zero vector would...
