Your classmate claims that the the function
f(x,y) = x^3(e^{x^2−3y^2})−(17−3x^2)(y−2)
does not have an absolute maximum nor an absolute minimum when the function is restricted to the disk x^2+y^2 ≤ 7.
Please explain why this is incorrect.
the domain of exponentials is all reals.
The domain of all polynomials is all reals.
So, the domain of f(x,y) is all reals
To determine whether the function f(x, y) has an absolute maximum or minimum on the disk x^2 + y^2 ≤ 7, we can follow these steps:
Step 1: Find the critical points of the function f(x, y) within the disk x^2 + y^2 ≤ 7.
To find the critical points, we need to solve the system of equations ∂f/∂x = 0 and ∂f/∂y = 0.
∂f/∂x = 3x^2e^(x^2 - 3y^2) + 2x(17 - 3x^2)(y - 2)
∂f/∂y = -6xy(e^(x^2 - 3y^2)) + (17 - 3x^2)
Step 2: Determine the points where the critical points lie within the disk.
Solve the equation x^2 + y^2 ≤ 7 to find the region within the disk.
Step 3: Evaluate the function f(x, y) at the critical points.
Substitute the values of x and y obtained from step 1 into the function f(x, y) to evaluate it at the critical points within the disk.
Step 4: Find the extreme values of f(x, y) within the disk.
Once we have the values of f(x, y) at the critical points, we can compare them to find the extreme values within the disk.
Step 5: Determine if there is an absolute maximum or minimum.
If there exists a point within the disk where f(x, y) has the greatest (maximum) or least (minimum) value compared to all other points, then the function has an absolute maximum or minimum, respectively. Conversely, if there is no such point, the function does not have an absolute maximum or minimum within the disk.
Now, let's apply these steps to the given function f(x, y) = x^3(e^(x^2-3y^2)) - (17 - 3x^2)(y - 2) and the disk x^2 + y^2 ≤ 7 to determine if the classmate's claim is incorrect.
Step 1: Find the critical points of f(x, y).
By calculating the partial derivatives, we can find the critical points of f(x, y) within the disk:
∂f/∂x = 3x^2e^(x^2 - 3y^2) + 2x(17 - 3x^2)(y - 2) = 0
∂f/∂y = -6xy(e^(x^2 - 3y^2)) + (17 - 3x^2) = 0
However, finding the explicit solutions to these equations is beyond the scope of this explanation.
Step 2: Determine the points within the disk.
By solving the equation x^2 + y^2 ≤ 7, we can find the region within the disk where the critical points lie.
Step 3: Evaluate f(x, y) at the critical points.
Substitute the values of x and y obtained by solving the system of equations into the function f(x, y) to evaluate it at the critical points within the disk.
Step 4: Find the extreme values of f(x, y) within the disk.
Compare the values of f(x, y) obtained from step 3 to find the extreme values within the disk.
Step 5: Determine if there is an absolute maximum or minimum.
If there is a point where f(x, y) has the greatest or least value compared to all other points within the disk, then the function has an absolute maximum or minimum, respectively. If there is no such point, the function does not have an absolute maximum or minimum within the disk.
Therefore, without explicitly solving the system of equations and evaluating the function at the critical points, it is not possible to definitively say whether the classmate's claim is incorrect.