Final answer:
The statement is false, as a zero or non-existent h'(c) indicates a critical point, but further tests are needed to determine if it's a relative maximum or minimum, or neither.
Step-by-step explanation:
The statement that at every point where h′(c) is zero or does not exist, h has a relative maximum or minimum is false. While it is true that the first part of the statement is a necessary condition for relative extrema, it is not a sufficient condition. A point where h′(c) is zero is called a critical point. However, a critical point can be a relative maximum, minimum, or neither (as in the case of a saddle point or inflection point).
To determine the nature of a critical point, one must use the First or Second Derivative Test. If h′(c) is zero and the second derivative at c is positive, h has a relative minimum at c. If h′(c) is zero and the second derivative at c is negative, h has a relative maximum at c. If the second derivative is zero or does not exist, other methods, like the Higher Derivative Test or graphing, may be necessary to classify the critical point.