Multivariable Extrema

Key idea

Use the Hessian to classify critical points.

In depth

Critical points have f_x = f_y = 0. Classify with the second derivative test using D = f_xx f_yy − (f_xy)²: D > 0 and f_xx > 0 gives a min, D > 0 and f_xx < 0 a max, D < 0 a saddle.

Worked example

f = x² − y²: critical point (0, 0), D = −4 < 0, so a saddle.

Real-world uses

• Finding the best combination of two inputs, such as temperature and pressure, in chemical plants

Common mistake

Using only f_xx and ignoring the discriminant.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Calculus

• Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
• Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
• Logarithmic Differentiation – Take logs before differentiating products and powers.
• Related Rates – Link rates of change of connected quantities.
• Linear Approximation – Approximate f near a using its tangent line.
• Differentials – Use dy = f′(x)dx for small changes.