The statements that could be false on the interval π/2 ≤ x ≤ 7π/4 are:
Statement (b): By the Extreme Value Theorem, there is a value c such that g(c) ≥ g(x) for π/2 ≤ x ≤ 7π/4.
Statement (d): By the Mean Value Theorem, there is a value c such that g'(c) = [g(7π/4) - g(π/2)] / (7π/4 - π/2).
How to analyze the statements given in terms of the properties mentioned.
The function g(x) = √(1 + cos(x)) is continuous on the interval π/2 ≤ x ≤ 7π/4.
a) By the Extreme Value Theorem, there is a value c such that g(c) ≤ g(x) for π/2 ≤ x ≤ 7π/4.
This statement aligns with the Extreme Value Theorem, which guarantees the existence of both maximum and minimum values for a continuous function on a closed interval.
Thus, it is true for g(x) within the given interval.
b) By the Extreme Value Theorem, there is a value c such that g(c) ≥ g(x) for π/2 ≤ x ≤ 7π/4.
This contradicts the Extreme Value Theorem, as g(x) cannot have both maximum and minimum values simultaneously on the interval.
Hence, this statement could be false.
c) By the Intermediate Value Theorem, there is a value c such that g(c) = (g(π/2) + g(7π/4)) / 2.
The statement involves the Intermediate Value Theorem, which ensures that for a continuous function on a closed interval, any value between f(a) and f(b) can be obtained for a point c between a and b.
Similarly, this statement is true for g(x) on the given interval.
d) By the Mean Value Theorem, there is a value c such that g'(c) = [g(7π/4) - g(π/2)] / (7π/4 - π/2).
The Mean Value Theorem asserts the existence of a point c where the instantaneous rate of change (derivative) of a function is equal to the average rate of change over the interval.
This statement seems to be using the Mean Value Theorem inappropriately because g(x) might not have a derivative at some points on the given interval. Thus, this statement could be false.
So, the statements that could be false on the interval π/2 ≤ x ≤ 7π/4 are:
Statement (b): By the Extreme Value Theorem, there is a value c such that g(c) ≥ g(x) for π/2 ≤ x ≤ 7π/4.
Statement (d): By the Mean Value Theorem, there is a value c such that g'(c) = [g(7π/4) - g(π/2)] / (7π/4 - π/2).