Question

Consider the polynomial p(x) = x^4 + 2x^3 - 3x^2 . Over the interval [-3 1] Determine all extreme points analytically. Employ the second derivative test to determine the maxima and minima. What are the coordinates of the global minimum?

Answer 1

To find extreme points of a polynomial, first find the derivative and set it to zero to find critical points. Use the second derivative test to determine if those points are minima or maxima. Assess these points and the endpoints of the interval to find the coordinates of the global minimum.

Step-by-step explanation:

To determine all extreme points analytically for the polynomial p(x) = x^4 + 2x^3 - 3x^2, we first find the derivative of the polynomial:

p'(x) = 4x^3 + 6x^2 - 6x.

Setting the derivative equal to zero gives us the critical points:

0 = 4x^3 + 6x^2 - 6x.

Factoring out x, we have:

0 = x(4x^2 + 6x - 6).

Using the quadratic formula or factoring further, we find the x-values of the critical points. Next, we check these points with the second derivative:

p''(x) = 12x^2 + 12x - 6.

For the second derivative test, if p''(x) > 0 at a critical point, it is a minimum, if p''(x) < 0, it is a maximum. We then evaluate each critical point within the interval [-3, 1] and also assess the endpoints of the interval since a global minimum or maximum can occur there as well. The coordinates of the global minimum are the x-value of the minimum critical point and the corresponding p(x).