Question

(a). Check directly where we can solve the equation F(x,y)=y²+y+3x+1=0 for y in terms of x.

(b). Check that your answer in part (a) agrees with the answer you expect from the implicit function theorem. Compute dy/dx.

Answer 1

Solve the given quadratic equation for y in terms of x and confirm the solution using the implicit function theorem by computing dy/dx.

Step-by-step explanation:

The student's question asks to solve the equation F(x,y) = y² + y + 3x + 1 = 0 for y in terms of x and to verify the solution using the implicit function theorem by computing dy/dx. To find y as a function of x, we must treat this equation as a quadratic in y and use the quadratic formula. After solving for y, we can then differentiate implicitly with respect to x to find dy/dx. The implicit function theorem tells us that if the partial derivative of F with respect to y is non-zero, then locally around any point where F(x,y)=0, there exists a function y(x) such that F(x, y(x))=0 and dy/dx can be computed as -F_x/F_y.