Question at position 1 The number of critical points of f(x,y)=x2+x2y+y2−2y+2 is31240

Question

Question at position 1 The number of critical points of f(x,y)=x2+x2y+y2−2y+2 is31240

BlackTom AI · Accepted Answer

We start by interpreting the problem: find the number of critical points of the function f(x,y) = x^2 + x^2 y + y^2 - 2y + 2.

First, compute the partial derivatives to locate critical points where the gradient is zero.

- f_x = ∂f/∂x = 2x + ∂(x^2 y)/∂x = 2x + 2xy = 2x(1 + y).
- f_y = ∂f/∂y = ∂(x^2 y)/∂y + ∂(y^2)/∂y + ∂(-2y)/∂y = x^2 + 2y - 2.

We need to solve the system: 2x(1 + y) = 0 and x^2 + 2y - 2 = 0.

Option A: If x = 0 from the first equation, substitute into the second: 0^2 + 2y - 2 = 0 ⇒ 2y = 2 ⇒ y = 1. This yields the critical point (0, 1).

Option B: If y = -1 from the first equation, substitute into the second: x^2 + 2(-1) - 2 = 0 ⇒ x^2 - 4 = 0 ⇒ x^2 = 4 ⇒ x = ±2. This gives two more critical points: (2, -1) and (-2, -1).

Now tally the distinct critical points found:
- (0, 1)
- (2, -1)
- (-2, -1)
Thus, there are three critical points in total.

Option-by-option assessment:
- Option 3: This choice aligns with the count we found (three critical points), but we will examine other options to understand why they do not fit.
- Option 1: If we guessed that there is only one critical point, this would ignore the branch where y = -1 leads to x = ±2, so it is inconsistent with the gradient equations.
- Option 2: Claiming only two critical points would miss either (0, 1) or one of (2, -1) or (-2, -1) derived from the systems above, making it incorrect.
- Option 4 or 0: Any assertion of four or zero critical points fails to satisfy the derived solutions from the system, as we explicitly found three distinct points.

In summary, solving the system from the gradient reveals three critical points, obtained via the two branches x = 0 and y = -1, leading to the coordinates (0, 1), (2, -1), and (-2, -1).

Question at position 1 The number of critical points of f(x,y)=x2+x2y+y2−2y+2 is31240

View Explanation

Log in for full answers

Similar Questions

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

Question at position 2 The function f(x,y)=13x3+12y2+xy−6x+3 has a relative minimum at(-2, 2)(-3, 3)(2, -2)(3, -3)(2, 2)

Question at position 2 The function f(x,y)=13x3+12y2+xy−6x+3f\left(x,y\right)=\frac{1}{3}x^3+\frac{1}{2}y^2+xy-6x+3 has a relative minimum at(-2, 2)(2, -2)(-3, 3)(2, 2)(3, -3)

Question at position 1 The number of critical points of f(x,y)=x2+x2y+y2−2y+2f\left(x,y\right)=x^2+x^2y+y^2-2y+2 is42301

f’(x) = 0 Therefore

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler