**1. ** Remember for a curve , we have maxima and minima occur whenever

(1)

What’s the multivariable analog to this notion?

**2. Definition.** If , we say is a **“Critical Point”** of .

Example 1.Consider . What are its critical points?

Solution: We find its gradient first

(2)

Next we need to set each component to vanish. This implies

(3)

is the only critical point.

**3. Problem:** How do we determine if a critical point describes a maxima or minima?

Lets consider the critical point for . We Taylor expand to second order about writing

(4)

where we use the matrix

(5)

Each row is precisely , and each column likewise is ; the intuition is .

Now, since we are Taylor expanding about a critical point, our approximation becomes

(6)

We can consider the behaviour of near by studying the properties of . Specifically, the signs of the eigenvalues tells us whether the critical point is a local maxima (all eigenvalues are positive) or a minima (all are negative) or some saddle point (mixture having both positive and negative eigenvalues). If there exists at least one eigenvalue that vanishes, this test is inconclusive.

**4. Parting Thoughts:** What if we want to optimize a function constrained to live on a surface?

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