determine whether the following statements are true and give an explanation or counterexample. a. the zeroes…

determine whether the following statements are true and give an explanation or counterexample. a. the zeroes of f are - 3, 1, and 4, so the local extrema are located at these points. choose the correct answer below. o a. false. a zero of f is a critical point and is a local extremum so long as f(x) changes sign. take, for example, the function f(x)=(x + 3)^3(x - 1)^3(x - 4)^3. o b. true. the zeros of f are local extrema so long as the denominator is nonzero at those points. o c. true. the zeros of f are the local extrema of f. o d. false. the zeros of f are the inflection points of f.

determine whether the following statements are true and give an explanation or counterexample. a. the zeroes of f are - 3, 1, and 4, so the local extrema are located at these points. choose the correct answer below. o a. false. a zero of f is a critical point and is a local extremum so long as f(x) changes sign. take, for example, the function f(x)=(x + 3)^3(x - 1)^3(x - 4)^3. o b. true. the zeros of f are local extrema so long as the denominator is nonzero at those points. o c. true. the zeros of f are the local extrema of f. o d. false. the zeros of f are the inflection points of f.

Answer

Explanation:

Step1: Recall critical - point and local - extrema relationship

A zero of the first - derivative (f') is a critical point. But a critical point is a local extremum if and only if the sign of (f'(x)) changes at that point.

Step2: Analyze the given statement

The statement claims that since the zeroes of (f') are (-3), (1), and (4), the local extrema are located at these points. This is false because just having a zero of (f') does not guarantee a local extremum. For example, consider the function (y=(x + 3)^3(x - 1)^3(x - 4)^3). The derivative (y') has zeroes at (x=-3), (x = 1), and (x = 4), but the sign of (y') does not change at these points, so they are not local extrema.

Answer:

A. False. A zero of (f') is a critical point and is a local extremum so long as (f'(x)) changes sign. Take, for example, the function (f(x)=(x + 3)^3(x - 1)^3(x - 4)^3).