Math 2.0

Real functions Theorem

Derivative vanishes at a local extremum

Builds on Local maximum and minimumPartial derivativeGradientDifferentiable function of several variables

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, let f ⁣:U→ℝf \colon U \to \mathbb{R}, and let a∈Ua \in U be a local extremum of ff. If the partial derivative ∂f/∂xi(a)\partial f / \partial x_i (a) exists for some ii, then ∂f/∂xi(a)=0\partial f / \partial x_i (a) = 0. In particular, if ff is differentiable at aa, then ∇f(a)=0\nabla f(a) = 0 and the derivative Df(a)Df(a) is the zero linear map.

Remarks

A point a∈Ua \in U where ff is differentiable and ∇f(a)=0\nabla f(a) = 0 is a critical point, or stationary point, of ff. The proof applies the one-variable theorem to t↦f(a+tei)t \mapsto f(a + t e_i), where eie_i is the ii-th standard basis vector. This function has a local extremum at t=0t = 0. The converse is false. For f(x,y)=x2−y2f(x, y) = x^2 - y^2, ∇f(0,0)=0\nabla f(0, 0) = 0, but (0,0)(0, 0) is a saddle point, not an extremum. The hypothesis that UU is open matters. On [0,1]2[0, 1]^2, f(x,y)=x+yf(x, y) = x + y has a maximum at (1,1)(1, 1), where ∇f=(1,1)\nabla f = (1, 1).

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Further reading

  1. Web https://en.wikipedia.org/wiki/Interior_extremum_theorem, Extension
About this page
Tag0F405477F8
URIm2://0F405477F8
Aliasesfermat-theorem-several-variables, interior-extremum-theorem-several-variables
Kindstatement, theorem
WritingW0 stub
MSC26B05
StatusThis copy of the site has no build data. Run m2 status 0F405477F8.
UsesLocal maximum and minimum, Partial derivative, Gradient, Differentiable function of several variables

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