Brandon was asked to identify the relative maximum of the polynomial function h(x) . Which answer choice identifies the correct value(s)?
(1 point)
Responses
There is a relative maximum at x=2.
There is a relative maximum at x equals 2 .
There is no relative maximum in this graph.
There is no relative maximum in this graph.
There is a relative maximum at x=0.
There is a relative maximum at x equals 0 .
There is a relative minimum at x=1.
There is no information provided about the polynomial function h(x), so it is not possible to determine the location of any relative maximum or minimum points. Therefore, the correct answer choice is: "There is no relative maximum in this graph."
To identify the relative maximum of a polynomial function, we need to locate the point(s) on the graph where the function reaches a local maximum.
In the given answer choices, the correct response for identifying the relative maximum of the polynomial function h(x) is:
There is a relative maximum at x=2.
To identify the relative maximum of a polynomial function, you need to find the critical points of the function and then analyze their nature.
The critical points occur where the derivative of the function equals zero or is undefined. To find these points, take the derivative of the polynomial function. Let's call the polynomial function h(x).
Once you have the derivative, set it equal to zero and solve for x. The values of x obtained by solving this equation will give you the critical points of the function.
Next, you need to determine whether these critical points are relative maximums, relative minimums, or neither. To do this, you can use the first or second derivative test.
To apply the first derivative test, analyze the sign changes of the derivative around each critical point. If the sign changes from positive to negative, the critical point is a relative maximum. If the sign changes from negative to positive, it is a relative minimum.
The second derivative test involves taking the second derivative of the function and evaluating it at the critical points. If the second derivative is positive, the critical point is a relative minimum. If it is negative, the critical point is a relative maximum.
Now, let's analyze the options provided:
1. "There is a relative maximum at x=2."
This option suggests that x=2 is a critical point and is a relative maximum. However, without further analysis, we cannot determine if this option is correct.
2. "There is no relative maximum in this graph."
This option suggests that there are no relative maximums in the graph. This cannot be determined without considering the critical points.
3. "There is a relative maximum at x=0."
This option suggests that x=0 is a critical point and is a relative maximum. Similarly to option 1, more analysis is needed to confirm if this is correct.
4. "There is a relative minimum at x=1."
This option suggests that x=1 is a critical point and is a relative minimum. Again, further analysis is required to confirm if this is correct.
To determine the correct answer choice, you need to find the critical points of the polynomial function h(x) and analyze their nature using the first or second derivative tests.