which of the following most accurately identifies the relative maximum and minimum of the polynomial…

Explanation:

Step1: Recall derivative - extrema relation

To find relative maxima and minima of a polynomial function, we set its first - derivative equal to zero and solve for (x) (critical points). Then we use the second - derivative test or analyze the sign change of the first - derivative around the critical points. But since we don't have the function (h(x)) itself, we assume we are to check the given points against the properties of relative extrema. A relative maximum occurs when the function changes from increasing to decreasing, and a relative minimum occurs when the function changes from decreasing to increasing.

Step2: Analyze options

We need to know the behavior of the function at the given (x) - values in each option. Without the function or its graph, we can't calculate directly. But we can eliminate options based on the fact that a non - constant polynomial function of degree (n\geq2) has both a relative maximum and a relative minimum (under appropriate conditions). Option 2 is incorrect because a non - constant polynomial usually has both types of extrema. For a polynomial, we expect to find both a maximum and a minimum in most cases. By checking the nature of extrema at the given (x) values in the remaining options, we assume we have some prior knowledge or a graph (not shown here) that would help us. If we assume we have analyzed the function's behavior around the points, we find that the correct relationship between relative maximum and minimum is given by the option where the function changes from increasing to decreasing at the maximum point and from decreasing to increasing at the minimum point.

Answer:

There is not enough information in the question to calculate the relative maxima and minima precisely. However, if we assume we have analyzed the function's behavior around the given points, based on the general properties of polynomial functions and the nature of relative extrema, we would choose the correct option among the ones provided. But as it stands, we can rule out the option that claims no relative minimum for a non - constant polynomial. If we had to make a guess without the function, we can't determine the answer with certainty. If we assume we have some external analysis (like a graph), we would pick the option that correctly describes the change in the function's behavior at the given (x) values. Since we can't calculate, we can't give a definite answer from the options. If we had the function (h(x)), we would:

1. Find (h'(x)) and set (h'(x)=0) to find critical points.
2. Use the second - derivative (h''(x)) or the first - derivative test (check the sign of (h'(x)) around the critical points) to determine if the critical points are relative maxima or minima. For example, if (h'(x)) changes sign from positive to negative at a critical point (x = c), then (h(c)) is a relative maximum. If (h'(x)) changes sign from negative to positive at (x = c), then (h(c)) is a relative minimum.

If we assume we have done the above analysis for the given function (not shown), and based on the general properties of polynomials, we note that a polynomial of degree (n\geq2) has a non - zero number of relative extrema. Option 2 is incorrect as a non - constant polynomial usually has both relative maxima and minima. Among the other options, we would choose the one that correctly describes the behavior of the function at the given (x) values. But without the function, we can't give a definite answer.

If we had to choose randomly among the remaining options (assuming we have no other information), we still can't be sure. But if we had the function (h(x)):

1. Differentiate (h(x)) to get (h'(x)).
   • Let (h(x)=a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_1x + a_0), then (h'(x)=na_nx^{n - 1}+(n - 1)a_{n - 1}x^{n - 2}+\cdots+a_1).
2. Set (h'(x) = 0) and solve for (x) to find the critical points.
3. Differentiate (h'(x)) to get (h''(x)).
4. Evaluate (h''(x)) at the critical points. If (h''(c)>0) at a critical point (x = c), then (h(c)) is a relative minimum. If (h''(c)<0) at (x = c), then (h(c)) is a relative maximum.

Since we don't have (h(x)), we can't perform these steps. So, we can't give a definite answer from the options provided.