If the graph of f " (x) is continuous and has a relative maximum at x = c, which of the following must be true? (5 points) the graph of f has an x-intercept at x = c the graph of f has an inflection point at x = c. the graph of f has a relative minimum at x = c none of the above is necessarily true.

The answer isnone of the above is necessarily true.the second derivative of function f  is just the rate of change of the first derivative. the first derivative of function f is just the rate of change of function f.in general, the derivatives of a function f cannot be used to determine the x-intercepts of the graph of function f. a rate of change does not tell you much about where a graph will intercept the y-axis, especially when we consider different vertical translations of f and how they do not affect f'. (consider y = x² + 1 and y = x² - 1. they both have the same derivative y' = 2x but different numbers of x-intercept.)we cannot guarantee that the graph of f has an inflection point at x = c. inflection points only occur when the second derivative, f'', changes from positive to negative or vice versa. a relative maximum on the graph of f'' would not result in a change in sign for f'', even if a point where a relative maximum of f'' occurs is on the x-axis.we cannot guarantee that the graph of f will have a relative minimum at x = c. information about the first derivative of f is needed to determine relative minimums on the graph of f.attached is an example that disproves the notion that any of those first three choices *must* be true. in my graph, the red graph of f'' is continuous has a relative maximum at x  = 0. you can see that the resulting graph of f in black (the equation for it is f(x) =  (-1/12)x⁴ + (3/2)x² + x - 1 ) does not have an x-intercept, inflection point, or relative minimum at x = 0. therefore, the three claims above are not necessarily true.

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