Explains why the following statements are true or false.

Question

a) On an open interval of x, if f’(x) < 0 and f”(x) > 0, then f is a decreasing function that is concave down.

b) if g’(3)=0, then g(x) is guaranteed to have a local minimum or maximum at x=3.

Answer 1

a) The first derivative of f(x) describes whether it is increasing or decreasing, and a negative value indicates a decreasing function, since the instantaneous slope is always negative. However, the second derivative describes the concavity of the original function, where a positive value is concave up. Therefore, this statement is false.

b) The First Derivative and Second Derivative Tests state that, in order to know whether g(x) has a relative maximum or minimum at x = 3, you must know that g'(3) = 0 AND either the change in the first derivative as it passes through x = 3 or the value of g''(3). Since neither of these are known, g(x) is not guaranteed to have a local maximum or minimum at x=3.