true or false, if f(7) = f(7) = 0, then x = 7 must be an inflection point. (remember to give your reasons in…

true or false, if f(7) = f(7) = 0, then x = 7 must be an inflection point. (remember to give your reasons in the written section. feel free to give examples to support your answer as well.) false true

true or false, if f(7) = f(7) = 0, then x = 7 must be an inflection point. (remember to give your reasons in the written section. feel free to give examples to support your answer as well.) false true

Answer

Explanation:

Step1: Recall inflection - point definition

An inflection point is a point where the second - derivative changes sign. Just because (f''(7)=0) does not mean it changes sign at (x = 7).

Step2: Provide a counter - example

Consider the function (f(x)=(x - 7)^4). First, find the first - derivative using the power rule ((u^n)^\prime=nu^{n - 1}u^\prime). Let (u=x - 7), then (f^\prime(x)=4(x - 7)^3). And (f^\prime(7)=4(7 - 7)^3 = 0). Second, find the second - derivative: (f''(x)=12(x - 7)^2). And (f''(7)=12(7 - 7)^2=0). But (f''(x)=12(x - 7)^2\geq0) for all (x), so the second - derivative does not change sign at (x = 7), and (x = 7) is not an inflection point.

Answer:

False