In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point.

Once you find a point where the gradient of a multivariable function is the zero vector, meaning the tangent plane of the graph is flat at this point, the second partial derivative test is a way to tell if that …

Oct 16, 2025 · This formula is called the Second Partials Test, and it can be used to classify the behavior of any function at its critical points, as long as its second partials exist there and as long as …

Oct 31, 2025 · Learn how to use the second partials test for identifying relative extrema and saddle points for critical points of surfaces with the 53rd lesson of Calculus 3 from JK Mathematics!

It should certainly be possible to tell which case we are dealing with by looking at the coe cients A, B, and C, and this is the idea behind the Second Partials Test.

Theorem (Second Partials Test). Suppose the second partial derivatives of f are continuous on a disk with center (a; b), and suppose that (a; b) is a critical point of f, i.e., rf(a; b) = ~0.

5 days ago · If f (x,y) is a two-dimensional function that has a local extremum at a point (x_0,y_0) and has continuous partial derivatives at this point, then f_x (x_0,y_0)=0...

(b) Use the second partials test to classify the critical points. Answer: (0,0) is a saddle point, the other two critical points are relative minima.

Once you find a point where the gradient of a multivariable function is the zero vector, meaning the tangent plane of the graph is flat at this point, the second partial derivative test is a way to tell if that …

Second Partials Test If f\p {x, y} f (x,y) is a twice differentiable function, then a point \p {a, b} (a,b) is a critical point if \nabla f\p {a, b} = \vec {0} ∇f (a,b) = 0.