Question

For the given polynomial, describe minima and maxima to the nea f(x)=x⁴-4x²+x+3

Answer 1

The student's question pertains to finding minima and maxima of a polynomial, which involves first and second derivatives of the function, but the provided information is not relevant to the usual mathematical procedure.

Step-by-step explanation:

The student is asking about describing the minima and maxima of the polynomial function f(x) = x⁴ - 4x² + x + 3. To find these, we would typically calculate the first derivative of the function, set it equal to zero, and solve for the values of x that are critical points. Then, we use the second derivative to determine the nature of each critical point, whether they are minima or maxima. However, the provided information does not align with the usual method of finding minima and maxima for a polynomial function. Instead, it seems to reference concepts related to potential energy and equilibrium in a physics context, and also includes an unrelated mention of a least-squares regression line. Without more relevant details, an accurate answer to the question cannot be given.

Answer 2

The minima and maxima of the polynomial function f(x) = x^4 - 4x^2 + x + 3 can be found using the first and second derivative tests, subsequently graphing the function to visualize the results.

Step-by-step explanation:

To find the minima and maxima of the polynomial function f(x) = x⁴ - 4x² + x + 3, we need to determine the critical points by taking the derivative of the function and setting it equal to zero. Once we have the critical points, the second derivative test can help us determine whether those points are minima, maxima, or points of inflection. To illustrate:

• First derivative: f'(x) = 4x³ - 8x + 1
• Set f'(x) = 0 and solve for x to find critical points.
• Use the second derivative, f''(x) = 12x² - 8, to evaluate each critical point.

If f''(x) is positive at a critical point, it's a minimum. If f''(x) is negative, it's a maximum. By applying this test, we can then graph the function to better visualize the minima and maxima and describe their nature.