The graph of f "(x) is continuous and decreasing with an x-intercept at x = 0. Which of the following statements is true?
The graph of f has a relative maximum at x = 0.
The graph of f has a relative minimum at x = 0.
The graph of f has an inflection point at x = 0.
The graph of f has an x-intercept at x = 0.
Thank you in advance!

An inflection point occurs when the second derivative changes signs. Since f"(x) is decreasing and has an x-intercept at 0, it changes signs from + to -, so there is an inflection point at x = 0.

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harshsuts054 harshsuts054 · Answer 2 · 2022-05-21T13:27:53+02:00

The statements "The graph of f has an inflection point at x = 0." is true

X-intercept

The meaning of X-INTERCEPT is the x-coordinate of a point where a line, curve, or surface intersects the x-axis.

How to solve this problem?

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 & there is a change in sign, the statement 'The graph of f has an inflection point at x = 0." is true.

From above reasons we can say that the statement "The graph of f has an x-intercept at x = 0." is true

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