7.1 Critical points and the second derivative test
2 min read•august 6, 2024
Finding critical points is key to understanding a function's behavior. We use partial derivatives to locate these points where the function might reach its highest or lowest values, or change direction.
The helps classify these critical points. By examining the , we can determine if a point is a , minimum, or , revealing the function's shape in that area.
Critical Points and Partial Derivatives
Identifying Critical Points
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Critical points occur where the partial derivatives of a multivariable function are simultaneously zero or undefined
To find critical points, set each equal to zero and solve the resulting system of equations
Critical points represent potential local maxima, local minima, or saddle points of the function
Gradient Vector and its Applications
The is a vector-valued function that points in the direction of of a scalar-valued function
Partial derivatives are the components of the gradient vector, representing the rates of change of the function with respect to each variable
The gradient vector is perpendicular to the or of the function at any given point
The magnitude of the gradient vector indicates the steepness of the function at a point (larger magnitude implies steeper slope)
Classifying Critical Points
Hessian Matrix and its Determinant
The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function
For a , the Hessian matrix is given by:
H(x,y)=[fxx(x,y)fyx(x,y)fxy(x,y)fyy(x,y)]
The of the Hessian matrix, denoted as det(H), helps classify critical points
If det(H)>0, the is either a local maximum or a
If det(H)<0, the critical point is a saddle point
Second Derivative Test for Classification
The second derivative test uses the Hessian matrix to classify critical points
For a critical point (a,b):
If det(H(a,b))>0 and fxx(a,b)<0, the point is a local maximum
If det(H(a,b))>0 and fxx(a,b)>0, the point is a local minimum
If det(H(a,b))<0, the point is a saddle point
A local maximum occurs when the function decreases in all directions from the critical point (peaks or hills)
A local minimum occurs when the function increases in all directions from the critical point (valleys or basins)
A saddle point occurs when the function increases in some directions and decreases in others from the critical point (resembles a saddle or mountain pass)