Is the given statement true or false? Justify your answer. If f''(c) =0, then (c, f(c)) is an inflection point of the curve y=f'(x)

Question

If the given statement true or false? Justify your answer.

If f''(c) =0, then (c, f(c)) is an inflection point of the curve y=f'(x)

Answer 1

The statement who's truth value is in question is a universal conditional statement and is true if it is true for every point in its domain. The statement is false if it is false for at least one point in its domain.

Our statement has as follows.

For all real function f(x), if f '' (c) = o , then c is an inflection point for the function f.

The answer is that the above statement is false since there exist a function such f(x) = x⁴ for which

f '' (0) = 0 and x= 0 is not an inflection point.

Answer 2

Depending on the function, the statement could be true or false.

For example, if f(x) = x³, then f"(0) = 0 and (0, 0) is an inflection point of f.

On the other hand, if f(x) = x², than f"(0) = 0, but (0,0) is not an inflection point.