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Second Derivative Test

The second derivative test says:

\[ f_x = 0 \text{ and } f_y = 0 \Rightarrow x=a, y=b \]
  • If at \((a,b)\) \(f_{xx} f_{yy} - (f_{xy})^2 > 0, f_{xx} > 0\), local min
  • If at \((a,b)\) \(f_{xx} f_{yy} - (f_{xy})^2 > 0, f_{xx} < 0\), local max
  • If at \((a,b)\) \(f_{xx} f_{yy} - (f_{xy})^2 < 0\), saddle point

Why?

We can use two-variable Taylor series expansion about \((a,b)\) to rewrite \(z = f(x,y)\) as:

\[ f(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) \] \[ + \frac{1}{2!} \left[ f_{xx}(a,b)(x-a)^2 + 2f_{xy}(a,b)(x-a)(y-b) + f_{yy}(a,b)(y-b)^2 \right] + \dots \]

(Note the first line is simply the tangent plane approximation of \(f(x,y)\))

At \((a,b)\), \(f_x(a,b) = f_y(a,b) = 0\) (definition of critical points). Let's truncate the series after the second-order terms (ignore the \(\dots\)).

\[ f(x,y) \approx f(a,b) + \frac{1}{2!} \left[ f_{xx}(a,b)(x-a)^2 + 2f_{xy}(a,b)(x-a)(y-b) + f_{yy}(a,b)(y-b)^2 \right] \]
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If \(f(a,b)\) is a local minimum, then

\[ f(x,y) - f(a,b) > 0 \]

So

\[ \frac{1}{2!} \left[ f_{xx}(a,b)(x-a)^2 + 2f_{xy}(a,b)(x-a)(y-b) + f_{yy}(a,b)(y-b)^2 \right] > 0 \]

or

\[ f_{xx}(a,b)(x-a)^2 + 2f_{xy}(a,b)(x-a)(y-b) + f_{yy}(a,b)(y-b)^2 > 0 \]

Let \(x-a = h, y-b = k\). We want

\[ f_{xx} h^2 + 2f_{xy} hk + f_{yy} k^2 > 0 \]

Divide by \(k^2\):

\[ f_{xx} \left(\frac{h}{k}\right)^2 + 2f_{xy} \left(\frac{h}{k}\right) + f_{yy} > 0 \]

Quadratic in the form of:

\[ f_{xx} w^2 + 2f_{xy} w + f_{yy} > 0 \]

Consider \(f_{xx} w^2 + 2f_{xy} w + f_{yy} = 0\) first. Its roots are:

\[ w = \frac{-f_{xy} \pm \sqrt{(f_{xy})^2 - f_{xx} f_{yy}}}{f_{xx}} \]

If \((f_{xy})^2 - f_{xx} f_{yy} < 0\) (or \(f_{xx} f_{yy} - (f_{xy})^2 > 0\)), then there are no real roots and \(f_{xx} w^2 + 2f_{xy} w + f_{yy}\) is always positive if...

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\[ f_{xx} > 0 \]

so, \[ f(x,y) - f(a,b) = \frac{1}{2!} \left[ f_{xx} h^2 + 2 f_{xy} hk + f_{yy} k^2 \right] > 0 \]

this is why \( D = f_{xx} f_{yy} - (f_{xy})^2 > 0 \) and \( f_{xx} > 0 \) means we have a local minimum at \( (a,b) \).

and if \( f_{xx} < 0 \), then as long as \( f_{xx} f_{yy} - (f_{xy})^2 > 0 \) means there are no real roots to \( f_{xx} w^2 + 2 f_{xy} w + f_{yy} = 0 \) and \( f_{xx} w^2 + 2 f_{xy} w + f_{yy} < 0 \)

If \( f_{xx} f_{yy} - (f_{xy})^2 < 0 \), then \( w \) will have real roots, and that means \( f_{xx} w^2 + 2 f_{xy} w + f_{yy} \) will sometimes be positive and sometimes be negative, so \( f(x,y) - f(a,b) \) will sometimes be greater than zero and sometimes less than zero.