Question about the gradient of a function

Question

We consider a function of class $C^1(\mathbb{R}^2)$, by searching a max or min between the points of $S=\{(x,y) \in \mathbb{R}^2 \;\; | \;\; x^2+y^2=1\}$, if $(x_0,y_0)$ is one of them, is it true that there exists $\lambda \in \mathbb{R}$ such that $\nabla f(x_0,y_0)=2\lambda (x_0,y_0)?$ Why?

Answer 1

Let $g(x,y)=x^2+y^2$. Then $S=\{(x,y)\in\Bbb R^1\mid g(x,y)=1\}$ and the method of Lagrange multipliers tells us that if $f$ has a local extreme at some point $(x_0,y_0)\in S$, then $\nabla f(x_0,y_0)=\lambda\nabla g(x_0,y_0)$, for some $\lambda\in\Bbb R$. And $\nabla g(x_0,y_0)=(2x_0,2y_0)$.

Answer 2

Assuming that you want the extremum of some $f(x,y)$, we can parameterize the locus as $x=\cos t,y=\sin t$.

Then by the chain rule, the stationary points of $f$ are given by

$$\frac{df}{dt}=\nabla f\cdot(-\sin t,\cos t)=0$$ and these are such that $$\nabla f=\lambda(\cos t,\sin t).$$

In other words, the derivative of the function when you remain in the locus is zero when the gradient is normal to the locus, i.e. parallel to a normal to the locus.

This is a quite general property.