Respuesta :
Statement D is false.
What is Continuous function?
A function is continuous when its graph is a single unbroken curve that you could draw without lifting your pen from the paper. That is not a formal definition, but it helps you understand the idea.
The Intermediate Value Theorem states that if a function f is continuous on an interval [a, b] and k is a value between f(a) and f(b), then there is at least one value c in the interval such that f(c) = k.
In this case, f is continuous on the interval [0, 4] and 5 is between f(0) = 2 and f(4) = 18, so there must be a value c such that f(c) = 5.
Therefore, statement A is true.
The Mean Value Theorem states that if a function f is differentiable on an interval [a, b] and continuous on the closed interval [a, b], then there exists a value c in the interval (a, b) such that f'(c) = (f(b) - f(a))/(b - a).
In this case, f is differentiable and continuous on the interval [0, 4], so there must be a value c such that f'(c) = (f(4) - f(0))/(4 - 0) = 4.
Therefore, statement B is true.
The Extreme Value Theorem states that if a function f is continuous on a closed interval [a, b], then f must attain both a maximum and a minimum value on that interval.
In this case, f is continuous on the interval [0, 4], so it must attain both a maximum and a minimum value on that interval.
However, the theorem does not specify whether the maximum or minimum value occurs at the endpoint or somewhere in the interior of the interval.
Therefore, statement C is false, and statement D is also false.
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