Question

Исследуйте на экстремум функцию:

Answer 1

Y=1/4x^4-2x^3+11/2x^2-6x+9/4
y'=x^3-6x^2+11x-6
y'=0

Answer 2

Finding the Extremum of a Function

To find the extremum of a function, we need to determine the critical points where the derivative of the function is equal to zero or does not exist. Let's explore the given function:

Function: Z = 8x^2 + 2xy - 5x + 6

To find the extremum of this function, we need to find the critical points by taking the partial derivatives with respect to x and y and setting them equal to zero.

Taking the partial derivative with respect to x:

∂Z/∂x = 16x + 2y - 5 = 0 Taking the partial derivative with respect to y:

∂Z/∂y = 2x = 0 Solving these equations simultaneously, we get:

16x + 2y - 5 = 0 (Equation 1) 2x = 0 (Equation 2)

From Equation 2, we find that x = 0. Substituting this value into Equation 1, we can solve for y:

16(0) + 2y - 5 = 0 2y = 5 y = 5/2

Therefore, the critical point is (x, y) = (0, 5/2).

To determine if this critical point is a maximum or minimum, we need to analyze the second-order partial derivatives. Taking the second partial derivatives:

∂^2Z/∂x^2 = 16 ∂^2Z/∂y^2 = 0 ∂^2Z/∂x∂y = 2 Using the second partial derivatives, we can calculate the discriminant:

D = (∂^2Z/∂x^2)(∂^2Z/∂y^2) - (∂^2Z/∂x∂y)^2 = (16)(0) - (2)^2 = -4

Since the discriminant is negative, we can conclude that the critical point (0, 5/2) is a saddle point.

Therefore, the function Z = 8x^2 + 2xy - 5x + 6 has a saddle point at (0, 5/2).