Graphs of f, f' and f"

Question

The graph of f ′(x) is continuous and decreasing with an x-intercept at x = 0. Which of the following statements is true? 

1. The graph of f has a relative maximum at x=0
2. The graph of f has a relative minimum at x=0
3. The graph of f has an inflection point at x=0
4. The graph of f has an x-intercept at x=0

Answer 1

The sign of the derivative will help you to determine where the original graph has relative minimums and maximums. It will not help you to determine inflection points, so the 3rd option is out. Since we do not know anything about coordinate for the original graph, we can also rule out the 4th option.

Since we are told that the graph is decreasing and constant with an x-intercept at 0, we know that it is positive before x=0 and negative after. That means that the original graph is increasing before x=0 and decreasing afterwards, making x=0 a relative maximum.