consider the funciton f(x)=4.5x-cosx+1 on the interval 0≤ x ≤1. the intermediate value theorem guarantees that there is a value c such that f(c)=N for which values of c and N? Fill in the follwoing mathmatical statements, giving an interval with non-zero length in each case.
For every N in __ ≤ N ≤ __ ,
there is a c __ ≤ c ≤ __
such that f(c) =N.
(a) f(0) ≤ f(1)
(b) pull the answer right from the text above.
For every N in 0 ≤ N ≤ 5.5,
there is a c 0 ≤ c ≤ 1
such that f(c) = N.
To find the values of c and N that satisfy the intermediate value theorem, we need to determine the interval in which f(c) takes on all possible values.
The given function is f(x) = 4.5x - cos(x) + 1 on the interval 0 ≤ x ≤ 1.
Step 1: Find the minimum and maximum values of f(x) on the interval [0,1].
To find the minimum and maximum values, we can analyze the behavior of the function and check its critical points.
Taking the derivative of f(x):
f'(x) = 4.5 + sin(x)
Setting f'(x) = 0 to find the critical points:
4.5 + sin(x) = 0
sin(x) = -4.5
Since the sin function has a maximum value of 1 and a minimum value of -1, we can conclude that there are no critical points within the interval [0,1]. Therefore, the minimum and maximum values of f(x) occur at the endpoints.
f(0) = 4.5(0) - cos(0) + 1 = 0 - 1 + 1 = 0
f(1) = 4.5(1) - cos(1) + 1 = 4.5 - cos(1) + 1
So, the minimum value of f(x) is 0 and the maximum value on the interval [0,1] is 4.5 - cos(1) + 1.
Step 2: Determine the interval for N.
Since the function reaches its minimum value at 0 and its maximum value at 4.5 - cos(1) + 1, the interval for N is: 0 ≤ N ≤ 4.5 - cos(1) + 1.
Step 3: Determine the interval for c.
The intermediate value theorem guarantees that for every N within the interval [0, 4.5 - cos(1) + 1], there exists a c within the interval [0, 1] such that f(c) = N.
So, the final mathematical statements are:
For every N in 0 ≤ N ≤ 4.5 - cos(1) + 1,
there is a c 0 ≤ c ≤ 1
such that f(c) = N.