How to use this calculator
Enter f using x, y and optionally z, and the point. Give a direction vector for the directional derivative (it is normalized for you) and a constraint for Lagrange multipliers. Critical points are found numerically inside the search box and classified.
Formulas
Gradient∇f = (fx, fy, fz)
Directional derivativeDuf = ∇f · u/|u|
Tangent planez = f(a, b) + fx(x − a) + fy(y − b)
Second-derivative testD = fxxfyy − fxy²
Lagrange∇f = λ∇g, g = c
Worked example
f = x²y + 3xy² at (1, 2), in the direction of (3, 4)
- fx = 2xy + 3y² = 16, fy = x² + 6xy = 13 at (1, 2).
- ∇f = (16, 13), |∇f| = 20.6.
- û = (0.6, 0.8), so Duf = 16(0.6) + 13(0.8) = 20.
- Tangent plane: z = 14 + 16(x − 1) + 13(y − 2).
Frequently asked questions
What does the gradient tell me?
It points in the direction of fastest increase, its length is that fastest rate, and it is perpendicular to the level curve through the point.
What if D = 0?
The second-derivative test is inconclusive; look at f along lines through the point or at higher-order terms.
Does Lagrange always give the maximum?
It gives the candidates. On a closed, bounded constraint the largest f among them is the maximum and the smallest the minimum.