Question

Find and classify all the critical points of the following function. (a) f(x,y)=7x−8y+2xy−x²+y³ (b) f(x,y)=x+2y⁴−lnx²y⁴

Answer 1

To find the critical points of a function, we need to find the partial derivatives of the function with respect to each variable and set them equal to zero. The critical points occur at the points where both partial derivatives are equal to zero.

Step-by-step explanation:

To find the critical points of a function, we need to find the partial derivatives of the function with respect to each variable and set them equal to zero. The critical points occur at the points where both partial derivatives are equal to zero. Let's find the critical points for the given functions:

(a) f(x,y) = 7x - 8y + 2xy - x^2 + y^3
Partial derivative with respect to x: 7 - 2x
Partial derivative with respect to y: -8 + 2x + 3y^2
Solving the equations 7 - 2x = 0 and -8 + 2x + 3y^2 = 0 gives us x = 7/2 and y = ±√(8/3)

(b) f(x,y) = x + 2y^4 - ln(x^2y^4)
Partial derivative with respect to x: 1 - 2/x
Partial derivative with respect to y: 8y^3 - 4y/x
Solving the equations 1 - 2/x = 0 and 8y^3 - 4y/x = 0 gives us x = 2 and y = 0