Final Answer:
The second derivative test cannot determine whether a critical point is a relative minimum or maximum for linear functions. This is due to their second derivative always being zero, making the test inconclusive for identifying extrema in linear functions. Thus the correct option is b) Linear functions.
Step-by-step explanation:
The second derivative test helps identify whether a critical point is a minimum, maximum, or inflection point based on the concavity of a function. For linear functions (option b), the second derivative is always zero, rendering the test inconclusive. Linear functions have a constant first derivative (slope) and a zero second derivative, making it impossible to determine maxima or minima using the second derivative test alone.
In contrast, quadratic functions (option a), cubic functions (option c), and trigonometric functions (option d) have changing concavities, allowing the second derivative test to ascertain the nature of critical points. Quadratic functions exhibit a constant second derivative, aiding in identifying extrema. Cubic functions possess varying concavity and can have inflection points or extrema determined by the second derivative. Trigonometric functions also display changing concavities, enabling the test to pinpoint maxima or minima.
However, linear functions defy this analysis since their second derivative is constantly zero, making it impossible to discern whether a critical point is a maximum, minimum, or an inflection point through the second derivative test. Hence, the second derivative test cannot conclusively identify extrema for linear functions.
Thus the correct option is b) Linear functions.