Second Derivative Test for Multivariable Functions

Overview

The second derivative test for f(x,y) classifies critical points using the Hessian matrix H = [[f_xx, f_xy],[f_xy, f_yy]]. The discriminant D = det(H) = f_xx*f_yy - (f_xy)^2 determines the outcome. For n variables, positive definiteness of H implies local minimum; negative definiteness implies local maximum; indefiniteness implies a saddle point.

Intuition

At a critical point, the function looks like a quadratic: ax^2 + bxy + cy^2. This bowl shape is a min (opens up in all directions) if the quadratic is positive definite; a cap (opens down) if negative definite; a saddle if neither. The discriminant D = ac - b^2 determines this: D > 0 means the same sign in all directions (bowl or cap); D < 0 means different signs (saddle).

Formal Definition

Definition

For f(x,y) with critical point (a,b): Hessian H = [[f_xx, f_xy],[f_yx, f_yy]] evaluated at (a,b). D = det(H) = f_xx*f_yy - f_xy^2. Classification: D > 0 and f_xx > 0 => local min; D > 0 and f_xx < 0 => local max; D < 0 => saddle point; D = 0 => test inconclusive. For n variables: H positive definite (all eigenvalues > 0) => local min; negative definite => local max.

H = \begin{pmatrix} f_{xx} & f_{xy} \\ f_{xy} & f_{yy} \end{pmatrix},\quad D = \det H = f_{xx}f_{yy} - f_{xy}^2

Hessian and discriminant

D > 0,\, f_{xx} > 0 \Rightarrow \text{local min};\quad D > 0,\, f_{xx} < 0 \Rightarrow \text{local max};\quad D < 0 \Rightarrow \text{saddle}

Classification

Notation

Notation	Meaning
$H_f$	Hessian matrix
$D = \det H_f$	Hessian determinant (discriminant)

Theorems

Theorem 1: Second Derivative Test (2D)

If (a,b) is a critical point of f and D = f_xx*f_yy - f_xy^2 at (a,b): D > 0 and f_xx > 0: local min. D > 0 and f_xx < 0: local max. D < 0: saddle point. D = 0: no conclusion (need higher-order analysis).

Worked Examples

1
f_x = 4x^3 - 4y = 0, f_y = 4y^3 - 4x = 0. So y = x^3 and x = y^3.
$y = x^3,\quad x = y^3 \Rightarrow x = x^9 \Rightarrow x(x^8-1)=0$
2
Critical points: (0,0), (1,1), (-1,-1).
3
Hessian: f_xx=12x^2, f_yy=12y^2, f_xy=-4. D=144x^2y^2-16.
4
At (0,0): D=0-16=-16<0. Saddle. At (1,1): D=144-16=128>0, f_xx=12>0. Local min. At (-1,-1): same. Local min.
$(\pm 1, \pm 1): \text{local min},; (0,0): \text{saddle}$

✓ Answer

(0,0) is a saddle; (1,1) and (-1,-1) are local minima.

Practice Problems

Mediumapplication

Find and classify all critical points of f(x,y) = x^2 + 2y^2 - 2x + 4y.

Common Mistakes

Common Mistake

Using f_xx or f_yy alone to classify without computing D.

Even if f_xx > 0, the critical point might be a saddle if D < 0. Both D and f_xx are needed for the classification.

Quiz

At a critical point where D = f_{xx}f_{yy} - f_{xy}^2 = 0, the test is:

InconclusiveAlways a saddle pointAlways a local extremumAlways a global min

Historical Background

The second derivative test for single-variable calculus (f''(a) > 0 means local min) generalizes to multiple variables using the Hessian matrix -- named after Ludwig Otto Hesse (1811-1874) who systematized second-order conditions in the 19th century. The conditions for local extrema involve the eigenvalues of the Hessian, with positive definiteness being the key criterion.

1844
Hesse introduces the determinant of second partial derivatives (Hessian matrix)
Ludwig Otto Hesse

Summary

Hessian H = [[f_xx, f_xy],[f_xy, f_yy]]. D = f_xx*f_yy - f_xy^2.
D > 0, f_xx > 0: local min. D > 0, f_xx < 0: local max. D < 0: saddle. D = 0: inconclusive.
For n variables: local min iff H positive definite (all leading minors > 0).

References

BookStewart, J. Multivariable Calculus. 8th ed. Brooks/Cole, 2015.