Question

If f is a continuous function on the closed interval [a,b]which of the following must be true?

a) There is a number c in the open interval (a,b) such thatf(c) = 0
b) There is a number c in the open interval (a,b) such thatf(a) < f(c) < f(b)
c) There is a number c in the closed intercal [a,b] such thatf(c) > or equal to f(x) for all x in [a,b]
d) There is a number c in the open interval (a,b) such thatf'(c)=0
e)There is a number c in the open interval (a,b) such thatf'(c) = (f(b)-f(a))/(b-a)
Explain your response and can you tell me if this is theintermediate value therom
and tell me which therom it is

Answer 1

The correct statement is option (c): There is a number c in the closed interval [a,b] such that f(c) is greater than or equal to f(x) for all x in [a,b].

Step-by-step explanation:

The correct statement for a continuous function f on the closed interval [a,b] is option (c): There is a number c in the closed interval [a,b] such that f(c) is greater than or equal to f(x) for all x in [a,b]. This statement is a direct result of the extreme value theorem, which states that a continuous function on a closed interval attains its maximum and minimum values.

Option (a) is not necessarily true because a continuous function can avoid passing through zero. Option (b) is not necessarily true because a continuous function can have its maximum or minimum value at the endpoints a or b. Option (d) is not necessarily true because the derivative of a continuous function can be nonzero at all points. And option (e) is not necessarily true because the derivative of a continuous function can be nonzero at all points as well.

Answer 2

The correct statement is option (c), which reflects the Extreme Value Theorem stating that a continuous function on a closed interval must attain a maximum and a minimum. Other options are not necessarily true without additional information about the function or its derivatives.

Step-by-step explanation:

If f is a continuous function on the closed interval [a,b], one statement that must be true is option (c): There is a number c in the closed interval [a,b] such that f(c) ≥ f(x) for all x in [a,b]. This statement is a direct consequence of the Extreme Value Theorem, which ensures that a continuous function on a closed interval must attain a maximum and a minimum value on that interval. Therefore, there exists at least one point c where f(c) is maximum (or equivalently, greater than or equal to any other value the function takes on the interval).

In contrast, statement (a) would require the existence of a root, which cannot be guaranteed without additional information about the values of f(a) and f(b). Statement (b) assumes a strict inequality without knowledge of how f(a) and f(b) relate to the other values of f(x). Statement (d) refers to the Mean Value Theorem but requires f to be differentiable, which is not given in the question. As for statement (e), it is an expression of the Mean Value Theorem, which states that for a continuous function that is also differentiable on the open interval (a,b), there exists some c in (a,b) such that f'(c) = (f(b)-f(a))/(b-a). However, differentiability is not guaranteed in this question, so this statement is not necessarily true either.