Find the local maxima, local minima, and saddle points, if any, for each function. z=y^2-6 y cos

Find the local maxima, local minima, and saddle points, if...

Key Concepts

Local Extrema

Local extrema refer to points in the domain of a function where the function takes on a maximum or minimum value relative to nearby points. Identifying local extrema is important for understanding the optimum values of functions in multivariable calculus.

Saddle Points

Saddle points are a type of critical point where the function does not achieve a local extremum. At a saddle point, the function bends upward in some directions and downward in others, making it a point of inflection in multiple dimensions.

Hessian Matrix

The Hessian matrix is the square matrix of second-order partial derivatives of a multivariable function. It plays a crucial role in determining the curvature of the function near a critical point and is used in the second derivative test for classifying the nature (local minima, maxima, or saddle points) of critical points.

Partial Derivatives

Partial derivatives are used to study functions of several variables by differentiating the function with respect to one variable while keeping the other variables constant. They form the basis for understanding the rate at which the function changes along each coordinate direction.

Critical Points

Critical points are points in the domain of a multivariable function where all first-order partial derivatives are zero or undefined. These points are candidates for local extrema or saddle points, making them essential for investigating the function’s behavior.

Second Derivative Test

The second derivative test utilizes the Hessian matrix to classify a critical point by analyzing the definiteness of the matrix. A positive definite Hessian indicates a local minimum, a negative definite Hessian indicates a local maximum, and an indefinite Hessian suggests a saddle point.

Transcript

00:01 For this problem, we were asked to find the local maxima, local minima, and saddle points, if any exist, for the function z equals y squared minus 6y cos of x plus 6.

00:09 So first we want to solve for when the partial derivatives will equal 0.

00:14 So partial derivative respect to x is going to be positive 6y cos sine of x equals 0.

00:22 And with respect to y, it will be 2y minus 6 cos x equals 0.

00:30 So we can see that to get the first to be 0, either we have y equals 0 or x equals some integer multiple of pi.

00:43 And then for this to be equal to 0, we would have to have that y equals 3 times cos of x.

00:53 So if essentially we have one case, if we have y equals zero, then we'd have to have that cos of x would need to equal zero.

01:03 So we'd have to have then that x is an odd multiple of pi by two.

01:09 So one set of cases would be x equals 2n plus 1 times pi by 2 and y equals 0, where n is an integer.

01:21 Then other case would be, well, let's see here.

01:25 We could have, if x equals n pi, we'd have that the sign of x equals zero, but we do need to be specific about is x going to be an odd multiple of pi or an even multiple of pi? so i'll write this as one set of points where x equals, write it as 2n plus 1 times pi, and in which case that would be giving us the odd multiples.

01:52 So that would be giving us then cose of an odd multiple of pi would be one second here.

02:03 Oh, actually, no, that is pretty clear.

02:05 An odd multiple of pi, that would be negative.

02:08 So we'd have that y equals negative three.

02:11 Or x equals 2n pi.

02:18 So that would be all of the even multiples.

02:20 So we'd get that y equals three at that point.

02:24 And we can calculate out some test values...