Consider the function f(x)=65x−cos(x)+2 on the interval 0 less than or equal to x less than or equal to 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 (1)is less than or equal to k is less than or equal to (?),
there is a c in (0) is less than or equal to c is less than or equal to (1)
such that .
I'm only stuck on the part with the question mark
To determine the values for the question marks in the mathematical statements, we need to consider the range of the given function f(x) = 65x - cos(x) + 2 on the interval [0, 1].
First, let's find the range of f(x). Since f(x) is a continuous function, we can find its minimum and maximum values on the closed interval [0, 1].
To find the minimum value, we consider the derivative of f(x) and set it equal to zero:
f'(x) = 65 + sin(x) = 0
sin(x) = -65
However, since sin(x) is bounded between -1 and 1, there is no value of x that satisfies sin(x) = -65. Therefore, the function does not have a minimum value on the interval [0, 1].
To find the maximum value, we consider the derivative of f(x) and find its critical points:
f'(x) = 65 + sin(x) = 0
sin(x) = -65
Again, sin(x) is bounded between -1 and 1, so there is no value of x that satisfies sin(x) = -65. Therefore, the function does not have a maximum value on the interval [0, 1].
Since the function f(x) does not have minimum or maximum values on the interval, its range is the set of all real numbers, i.e., (-∞, +∞). Therefore, for every k such that k is a real number, there exists a c in the interval (0, 1) such that .
In summary,
For every k in (-∞, +∞),
there is a c in (0, 1) such that .