Question

True or False? In Exercises 55-60, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is increasing on \([a, b],\) then the minimum value of \(f(x)\) on \([a, b]\) is \(f(a)\)

Short answer

The statement is True, assuming the function is defined and continuous on the interval [a, b].

Step-by-step solution

Understand the statement

The given statement claims that for any function f that is increasing on an interval [a, b], the minimum value of f(x) on that interval is reached at x = a i.e., f(a).

Analyze the statement

It helps to note that the function f is assumed to be defined and continuous on the closed interval [a, b]. If this is the case, then by definition of an increasing function, for any two numbers x and y in [a, b], if x < y, then f(x) < f(y). Thus, considering the endpoints of the interval, at x = a, we will have f(a) ≤ f(x) for any x in [a, b]. This means that the minimum value of f(x) is indeed at x = a.

Provide the final answer

Considering the definition of an increasing function, and assuming that the function is continuous and defined on the interval [a, b], the statement is True.