Decide if the statement is true or false. give an explanation for your answer. let f(x) = [x], the largest integer less than or equal to x. then f'(x) = 0, so f(x) is constant by the constant function theorem.

a. true. this function satisfies the constant function theorem because f'(x) = 0 everywhere.
b. false. this function does not satisfy the constant function theorem because the derivative f'(x) is not equal to zero everywhere, and the function is not continuous at integral values of x, so f'(x) does not exist there.

1. Since the function f(x)= [x] is not continuous on integer values, its derivative undefined at the integers. This mean that derivative us not defined.

Now, we the constant value function states that :- Suppose that f is continuous on[a,b] and differentiable on(a,b). If f'(x)= 0 on (a,b)then f is constant on [a,b].

Since, f(x) is not continuous and its derivative is undefined. Hence, it doesn't follow the constant value theorem.

Thus, the given statement is FALSE

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