Which of the following are critical points of the function [math: x3−3x2y+3y2]?

Question

BlackTom AI · Accepted Answer

Compute the partial derivatives: f_x = 3x^2 − 6xy = 3x(x − 2y) and f_y = −3x^2 + 6y = 3(2y − x^2). Critical points occur where both partials are zero.

1) Relevant knowledge points and concepts
- A critical point for a function of two variables is any point where the gradient (both partial derivatives) is zero.
- Solve the system f_x = 0 and f_y = 0 simultaneously.

2) Step-by-step reasoning
Set the partials to zero:
- f_x = 3x(x − 2y) = 0 implies x = 0 or x − 2y = 0 (i.e. y = x/2).
- f_y = 3(2y − x^2) = 0 implies 2y − x^2 = 0 (i.e. y = x^2/2).

Solve the two cases:

Case A: x = 0.
- Substitute into f_y condition: 2y − 0 = 0 ⇒ y = 0.
- Gives critical point (0, 0).

Case B: y = x/2.
- Substitute into f_y condition y = x^2/2 ⇒ x/2 = x^2/2.
- Multiply both sides by 2: x = x^2 ⇒ x^2 − x = 0 ⇒ x(x − 1) = 0.
- Solutions: x = 0 (gives y = 0 again) or x = 1. For x = 1, y = 1/2.
- Gives critical point (1, 1/2).

3) Intermediate calculations shown above:
- f_x = 3x^2 − 6xy = 3x(x − 2y)
- f_y = −3x^2 + 6y = 3(2y − x^2)
- Solving produced (0,0) and (1,1/2).

4) Why the correct answer is correct
Both (0,0) and (1,1/2) make f_x = 0 and f_y = 0 simultaneously, so they are critical points by definition.

5) Why the other options are incorrect
- (0.5, 0.5): does not satisfy 2y = x^2 (2·0.5 = 1 but x^2 = 0.25), nor y = x/2 (0.5 ≠ 0.25).
- (0.5, 1): fails both equations (2·1 = 2 but x^2 = 0.25).
- (1,1): y = 1 does not equal x/2 = 0.5 nor x^2/2 = 0.5.
Thus these do not make both partials zero.

6) Conclusion
The critical points are (0,0) and (1, 0.5), matching options a and e.

Which of the following are critical points of the function [math: x3−3x2y+3y2]?多项选择题

类似问题

Which of the following are critical points of the function \(x^3-3x^2y+3y^2\)?

Let \(f(x) = x^{3} + 5 x^{2} - 5 x + 2.\) Which of the following is a correct classification of the critical points of \(f\)?

Let 𝑓 ( 𝑥 ) = 𝑥 3 − 5 𝑥 2 + 5 𝑥 + 2. Which of the following is a correct classification of the critical points of 𝑓 ?

Let 𝑓 ( 𝑥 ) = 𝑥 3 − 5 𝑥 2 + 5 𝑥 + 2. Which of the following is a correct classification of the critical points of 𝑓 ?

For what values of 𝑥 does the function 𝑦 = 2 𝑥 3 + 9 𝑥 2 + 12 𝑥 + 6 have a gradient of zero?

Determine all critical points for the function. f(x) = (x - 2)7

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单