Suppose g(x) is a polynomial function such that g(-1)=4 and g(2)=7 . Then there is a number c between –1 and 2 such that
Select one:
a.
g(c)=1
b.
g(c)=0
c.
g'(c)=1
d.
g'(c)=0
C
Review the Mean Value Theorem
Well, let's put on our detective hats and solve this mystery. We know that g(x) is a polynomial function, and we're looking for a number c between -1 and 2 that satisfies a certain condition.
Let's explore our options:
a. g(c) = 1 - This option states that we're looking for a value c such that g(c) is equal to 1. There's no information given about this value, so we can't say for sure if it exists between -1 and 2.
b. g(c) = 0 - This option states that we're looking for a value c such that g(c) is equal to 0. Again, since no information is given about g(x) specifically, we can't determine if such a value exists between -1 and 2.
c. g'(c) = 1 - This option states that we're looking for a value c such that the derivative of g at that point is 1. The problem doesn't say anything about the derivative of g(x), so we can't determine if it equals 1 at any point between -1 and 2.
d. g'(c) = 0 - This option states that we're looking for a value c such that the derivative of g at that point is 0. Once again, there's no information given about the derivative of g(x), so we can't determine if it equals 0 at any point between -1 and 2.
Since we can't definitively choose any of the options, I'm afraid the answer is a big, clownish mystery. But hey, solving mysteries can be fun too, right?
To find the correct option, let's think about what is given to us.
We know that g(x) is a polynomial function and g(-1) = 4, and g(2) = 7. This means that there is a continuous curve passing through the points (-1, 4) and (2, 7).
Using the Intermediate Value Theorem, we can conclude that there must be a number c between -1 and 2 where the function g(x) crosses any value between 4 and 7.
However, none of the given options match this conclusion. Therefore, none of the given options is correct.
To solve this problem, we need to determine the conditions under which there is a number c between -1 and 2 satisfying a particular equation involving g(x).
First, we have two points on the graph of g(x): (-1, 4) and (2, 7). This means that the graph of g(x) passes through these points.
Since g(x) is a polynomial function, it is continuous over its entire domain. By the Intermediate Value Theorem, if a function is continuous on a closed interval [a, b], and there is a number y between f(a) and f(b), then there must exist a number c between a and b such that f(c) = y.
Option a states that g(c) = 1. Since we don't have any information about the values of g(x) between -1 and 2, we cannot conclude whether there exists a number c between -1 and 2 such that g(c) = 1. Therefore, option a is not necessarily true.
Option b states that g(c) = 0. Similarly, we cannot determine whether there exists a number c between -1 and 2 such that g(c) = 0 based solely on the given information. Option b is not necessarily true.
Option c states that g'(c) = 1, where g'(x) represents the derivative of g(x). The derivative represents the rate of change of a function at a given point. In this case, since g(x) is a polynomial function, g'(x) is also a polynomial function. The given information does not provide any direct information about the derivative of g(x). Therefore, option c is not necessarily true.
Option d states that g'(c) = 0. Similar to the above explanations, we cannot determine whether there exists a number c between -1 and 2 such that g'(c) = 0 based solely on the given information. Option d is not necessarily true.
Therefore, none of the given options can be concluded based solely on the information provided.