Question

Which of the following statements about the function (f), if true, cannot be used to conclude that (f) is de...

a) (f''(x) > 0) for all (x)
b) (f'(x) = 0) at a local miniμm
c) (f'(x) = 0) at a local maxiμm
d) (f'(x) = 0) for all (x)

Answer 1

The correct statement that cannot be used to conclude that the function is decreasing is (d) (f'(x) = 0) for all (x), because it indicates a constant function, which is neither increasing nor decreasing.

Step-by-step explanation:

The question concerns understanding the characteristics of a function based on its derivatives. Specifically, we need to identify which statement, if true, cannot be used to conclude that the function is decreasing.

(f''(x) > 0) for all (x) indicates the function is concave up everywhere, but does not necessarily mean the function is increasing or decreasing. (f'(x) = 0) at a local minimum or maximum suggests that the slope of the tangent line at that point is zero, representing a horizontal tangent line at those points. However, neither condition specifies anything about the increasing or decreasing nature of the function beyond those points. On the other hand, (f'(x) = 0) for all (x) implies that the function has a constant value; hence it is neither increasing nor decreasing, but it is constant.

The correct answer is d) (f'(x) = 0) for all (x), which indicates a constant function.