Overview
Finding maxima and minima of f(x,y) involves finding critical points where both partial derivatives are zero, then using the second derivative test (with the Hessian determinant) to classify each point as a maximum, minimum, or saddle point.
Imagine a mountainous landscape. Peaks are maxima, valleys are minima, and mountain passes (where you go up in one direction but down in another) are saddle points. The second derivative test is your way of distinguishing between them.
Maxima & Minima of Two Variables
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Mathematical Foundation
Formal Statement
Required Conditions
- • First, find critical points by solving fₓ = 0 and f_y = 0 simultaneously
- • If D > 0 and fₓₓ < 0 → local maximum
- • If D > 0 and fₓₓ > 0 → local minimum
- • If D < 0 → saddle point
- • If D = 0 → test is inconclusive
Common Mistakes & Pitfalls
- ⚠ Forgetting to check the sign of fₓₓ along with D — D > 0 alone doesn't tell you max vs min.
- ⚠ Missing critical points by not solving the system of equations completely.
- ⚠ Confusing saddle points with inconclusive results (D < 0 vs D = 0).