Calculus

The graph of f ′(x) is continuous and decreasing with an x-intercept at x = 0. Which of the following statements is false? (4 points)

The graph of f has an inflection point at x = 0.

The graph of f has a relative maximum at x = 0.

The graph of f is always concave down.

The graph of the second derivative is always negative.

I know that the first two are right, but not sure about the second two.

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  1. since f' is decreasing, we know that f" < 0

    So, at x=0, the slope is 0, and the curve is concave down, so it is a max there.

    if f" is always negative, the curve is always concave down.

    Looks like they're all true.

    Consider the parabola y = -x^2

    f' = -2x, which is always decreasing, and f'(0) = 0

    f" = -2, and the parabola is always concave down.

    So, only #1 is false.

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