Consider the function f(x)=8.5x−cos(x)+2 on the interval 0¡Üx¡Ü1 . The Intermediate Value Theorem guarantees that there is a value c such that for which values of c and k? Fill in the following mathematical statements, giving an interval with non-zero length in each case.
For every k in ______¡Ük¡Ü______
there is a c in 0 ¡Üc¡Ü 1
What is "c in 0 ¡Üc¡Ü 1" ?
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Consider the function f(x) = 8.5 x − cos(x) + 2 on the interval 0 ¡Ü x ¡Ü 1. The Intermediate Value Theorem guarantees that there is a value c such that for which values of c and k? Fill in the following mathematical statements, giving an interval with non-zero length in each case.
For every k in ____ ¡Ü k ¡Ü___ ,
there is a c in 0 ¡Ü c ¡Ü 1
such that .
To find the values of c and k that satisfy the given conditions, we need to determine the interval for which the Intermediate Value Theorem holds. The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and k is any value between f(a) and f(b), then there exists a value c in the interval (a, b) such that .
In this case, we have the function f(x) = 8.5x - cos(x) + 2 defined on the interval 0 ≤ x ≤ 1. To apply the Intermediate Value Theorem, we need to determine the range of f(x) over this interval.
First, let's find the range of the function on the interval [0, 1]. We can start by finding the minimum and maximum values of f(x) on this interval. Since the function includes a trigonometric term, we know that the cosine function has a maximum value of 1 and a minimum value of -1.
To find the maximum and minimum values of f(x), we can consider the extreme values of each term separately. The term 8.5x is an increasing linear function, and its minimum occurs at x = 0 and its maximum occurs at x = 1. Therefore, the minimum value of 8.5x on [0, 1] is 8.5(0) = 0, and the maximum value is 8.5(1) = 8.5.
The term -cos(x) ranges from -1 to 1, so the minimum value of -cos(x) on [0, 1] is -1 and the maximum value is 1.
Finally, adding the terms together, we have the range of f(x) on the interval [0, 1] as:
0 - 1 + 2 ≤ f(x) ≤ 8.5 + 1 + 2
1 ≤ f(x) ≤ 11.5
Now we can determine the values of c and k for which the Intermediate Value Theorem holds. From the range, we observe that for every value of k within the interval [1, 11.5], there exists a value c in the interval [0, 1] such that .
Therefore, the values of c and k that satisfy the given conditions are:
For every k in 1 ≤ k ≤ 11.5,
there exists a c in 0 ≤ c ≤ 1 such that .