To find the local maxima, local minima, and saddle points for the function f(x, y) = x³ + 3y² + 3xy – 3y, we first need to compute the critical points.
We begin by calculating the first order partial derivatives:
- fx(x, y) = 3x² + 3y
- fy(x, y) = 6y + 3x – 3
Next, we set these derivatives equal to zero:
- 3x² + 3y = 0 (1)
- 6y + 3x – 3 = 0 (2)
From (1), we can express y in terms of x: y = -x².
Substituting this into (2) gives us:
6(-x²) + 3x – 3 = 0
This simplifies to:
-6x² + 3x – 3 = 0
Multiplying through by -1, we get:
6x² – 3x + 3 = 0.
This quadratic can be solved using the quadratic formula:
x = [ -(-3) ± √((-3)² – 4 * 6 * 3) ] / (2 * 6).
Calculating the discriminant:
D = 9 – 72 = -63
Since the discriminant is negative, there are no real solutions for x. Thus, there are no critical points in the real plane.
As there are no critical points, the function does not have any local maxima or minima. However, we can further explore the nature of the function.
To understand the nature of the function, we may analyze the behavior as x and y approach infinity or negative infinity. The leading term x³ suggests that the function diverges to infinity as x approaches infinity in the positive direction. In both negative and positive directions, local behaviors led to the conclusion of no local extrema detected.
Finally, since we have no critical points, we conclude that there are no local maxima, local minima, or saddle points for the function.