Question

The function f is defined for all x in the closed interval [a,b]. If f does not attain a maximum value on [a,b], which of the following must be true?

answer choices
O f is not continuous on [a,b].

O f is not bounded on [a,b].

O f does not attain a minimum value on [a,b].

O The graph of f has a vertical asymptote in the interval [a,b].

O The equation f'(x) = 0 does not have a solution in the interval [a,b].

Answer 1

The function f does not attain a maximum value on [a,b], indicating that it may be unbounded within that interval, which aligns with the Extreme Value Theorem that requires bounded, continuous functions to have maximum and minimum values on a closed interval.

Step-by-step explanation:

If the function f is defined for all x in the closed interval [a,b] and does not attain a maximum value on [a,b], then one possibility is that f is not bounded on [a,b]. By the Extreme Value Theorem, if a function is continuous on a closed interval, it must attain both a maximum and minimum value on that interval. Thus, if f does not attain a maximum value, it might either be discontinuous or not bounded within the interval. However, the only option that deals directly with the boundness is f is not bounded on [a,b]. The existence of vertical asymptotes, non-existence of critical points, or not attaining a minimum value are not necessarily implicated by the fact that f does not have a maximum on the interval.

Answer 2

The correct option is b.

If a function f on a closed interval [a,b] does not attain a maximum value, it implies that f is not bounded on [a,b], due to the Extreme Value Theorem.

Step-by-step explanation:

If a function f defined on a closed interval [a,b] does not attain a maximum value, one possibility that must be true is that f is not bounded on [a,b]. This is due to the Extreme Value Theorem, which states that a continuous function on a closed interval must attain a maximum and minimum value. If f does not attain a maximum, it violates this theorem, suggesting that the function is unbounded.

However, it does not necessarily mean that f is not continuous, has a vertical asymptote, does not attain a minimum value, or that the equation f'(x) = 0 does not have a solution in the interval. Those possibilities can occur under different circumstances or assumptions for function f.