Use a graph or level curves to find the local maximum and minimum values and saddle points of the function. Then use calculus to find these values precisely. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x,y)=sin(x)+sin(y)+sin(x+y)+4, 0≤x≤2π, 0≤y≤2π
I don't know how to find these values precisely.
You need ∂f/∂x=0 and ∂f/∂y=0
∂f/∂x = cosx+cos(x+y)
∂f/∂y = cosy+cos(x+y)
clearly, this holds at (π,π), which is a saddle point
The max/minima will have to be found using some numeric technique, but I get
max at (1.0472,1.0472)
min at (-1.0472,-1.0472)
okay, why does it says use calculus to find these values precisely? How do I do that?
To find the local maximum and minimum values and saddle points of the function f(x,y) = sin(x) + sin(y) + sin(x+y) + 4, we can follow these steps:
Step 1: Plot the level curves:
We can plot the level curves of the function to get an idea of the behavior of the function. The level curves are curves on which f(x,y) takes the same value. We can use a graphing software or online tool to plot the level curves.
Step 2: Analyze the level curves:
By analyzing the level curves, we can identify the locations of local maximum and minimum values and saddle points. At the critical points, the gradient (∇f) of the function will be zero or undefined.
Step 3: Use calculus to find precise values:
To find the precise values, we need to find the critical points and then use the second derivative test to determine the nature of each critical point (whether it is a maximum, minimum, or saddle point).
Let's start with finding the critical points of the function:
Step 4: Find the partial derivatives:
To find the critical points, we need to take partial derivatives of f(x,y) with respect to x and y.
∂f/∂x = cos(x) + cos(x+y)
∂f/∂y = cos(y) + cos(x+y)
Step 5: Set the partial derivatives equal to zero:
To find the critical points, we set the partial derivatives equal to zero and solve the resulting equations.
cos(x) + cos(x+y) = 0 ----(1)
cos(y) + cos(x+y) = 0 ----(2)
Now, solving equations (1) and (2) simultaneously may be challenging. Let's simplify it further.
Step 6: Combine the equations:
Adding equations (1) and (2), we have:
cos(x) + cos(x+y) + cos(y) + cos(x+y) = 0
2cos(x+y) + cos(x) + cos(y) = 0
Step 7: Simplify the equation further:
Using the trigonometric identity cos(A + B) = cos(A)cos(B) - sin(A)sin(B), we can rewrite the equation as:
2cos(x)cos(y) - 2sin(x)sin(y) + cos(x) + cos(y) = 0
Step 8: Rearrange the equation:
Rearranging the terms, we get:
2cos(x)cos(y) + cos(x) + cos(y) = 2sin(x)sin(y)
Step 9: Simplify the equation further:
Using the trigonometric identity sin(2A) = 2sin(A)cos(A), we can rewrite the equation as:
cos(x)(2cos(y) + 1) = sin(x)(2sin(y))
Step 10: Solve the trigonometric equation:
To solve the equation, we can divide both sides by sin(x)cos(x) and obtain:
(2cos(y) + 1) / sin(y) = 2 / tan(x)
Now, we have two separate equations:
(2cos(y) + 1) / sin(y) = 2 / tan(x) ----(3)
(2cos(x) + 1) / sin(x) = 2 / tan(y) ----(4)
From equations (3) and (4), we can solve for x and y, respectively.
By solving these equations precisely, we can find the critical points and then perform the second derivative test to determine the nature of each critical point. However, solving these equations algebraically is quite challenging, if not impossible. Hence, we need to resort to numerical methods or approximation techniques to find the precise values of the critical points.
To find the local maximum and minimum values, saddle points, and their coordinates for the function f(x, y) = sin(x) + sin(y) + sin(x+y) + 4 over the given domain, we can follow these steps:
Step 1: Use a graph or level curves to visually determine the approximate location of the extrema and saddle points.
Step 2: Use calculus to find the exact values of these points.
In this case, we will start by using a graph or level curves to visualize the function and get an idea of where the critical points might be located.
Step 1: Visualizing the function using level curves:
To graphically analyze the function, we need to plot some level curves. Level curves are curves in the xy-plane along which the function has constant values.
Since the range of x and y is from 0 to 2π, we will create a grid with x and y values ranging from 0 to 2π. Then, for different values of the constant, we will plot the level curves.
Using a graphing calculator or software, we can plot the level curves. The graph should show the function f(x, y) = sin(x) + sin(y) + sin(x+y) + 4 as a surface with curves along which the function has constant values.
Step 2: Using calculus to find the precise values:
To find the local maximum, minimum, and saddle points of the function precisely, we need to find its partial derivatives with respect to x and y, and set them equal to zero. The critical points occur at these locations.
1. First, find the partial derivative with respect to x:
∂f/∂x = cos(x) + cos(x+y)
2. Second, find the partial derivative with respect to y:
∂f/∂y = cos(y) + cos(x+y)
3. To find critical points, set both partial derivatives equal to zero:
cos(x) + cos(x+y) = 0, and
cos(y) + cos(x+y) = 0
4. Solve these two equations simultaneously to find the values of x and y at the critical points. You can use algebraic methods or numerical methods to solve these equations. For example, you may use algebraic manipulation or a numerical solver like Newton's method or gradient descent.
Once you have the critical points, you can classify them by using the second-derivative test or Hessian matrix. Evaluating the second partial derivatives and using the tests will help you determine whether each point is a local maximum, minimum, or saddle point.
By following these steps, you should be able to find the local maximum and minimum values and saddle points precisely for the given function.