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

Question

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

A	The graph of f has a relative maximum at x = 0.
B	The graph of f has a relative minimum at x = 0.
C	The graph of f has an inflection point at x = 0.
D	The graph of f has an x-intercept at x = 0.

Answer 1

F''(x) tells you whether the function is concave up, down, or where it's inflection points are.

F"(x)>0 tells you it's concave up.

F"(x)<0 tells you it's concave down.

F"(x)=0 tells you there's a possible point of inflection at x.

Since this function is continuous and decreasing, we can assume that before it intercepts x at 0, it is concave up and after it intercepts, it continues to decrease which makes it concave down. Since F"(x)=0 AND there is a change in sign, the statement 'The graph of f has an inflection point at x = 0." is true.