The critical points of the function f(x, y) = x³ + y³ + 3x² + 6y² - 9x + 9y + 1 are determined by solving the partial derivatives. The first derivative with respect to x, df/dx = 3x² + 6x - 9, yields critical points at x = -3 and x = 1. The first derivative with respect to y, df/dy = 3y² + 12y + 9, results in critical points at y = -3 and y = -1. The corresponding critical points in the xy-plane are (-3, -3), (-3, 1), (1, -3), and (1, 1).
PREREQUISITESStudents and professionals in mathematics, particularly those studying calculus and optimization, will benefit from this discussion. It is also useful for anyone looking to understand the process of finding critical points in multivariable functions.
fx = 0 when x = -3 or x = 1simba_ said:Homework Statement
Find the critical points of
x3 + y3 + 3x2 + 6y2 - 9x + 9y +1
you do not need to define the critical points
Homework Equations
The Attempt at a Solution
i have
df/dx = 3x2 + 6x - 9 and when i solve this x = -3, 1
but i don't know what the corresponding y values are
df dy = 3y2 + 12y + 9 and so y = -3, -1
and here i don't know what the corresponding x values are