Final answer:
To choose the correct inequality without evaluating the definite integrals, properties of definite integrals and area under the curve in probability contexts are used. Option (a) can generally be assumed true, while options (c) and (d) would require further information about f(x), and option (b) cannot be assured without more context.
Step-by-step explanation:
In order to determine which inequality would make the statement true without evaluating the definite integrals, we need to apply our understanding of the properties of definite integrals and the concept of area under a curve in the context of a continuous probability distribution. With the information provided, which hints at the curve of a probability density function, we will consider the integral of f(x) over various intervals and what it represents in terms of probability.
For option (a), ∫0 to 1 f(x)dx ≥ 0, this is likely to be true since in a probability density function, the area under the curve representing the probability cannot be negative.
The statement for option (b), ∫1 to 2 f(x)dx = 0, implies there is no area under the curve between x=1 and x=2, which might indicate that f(x) is zero in this interval. However, without more information about the function f(x), we cannot be sure that this is the case for the entire interval, so we cannot confidently declare this statement as always true.
For option (c), ∫2 to 3 f(x)dx < 0, This suggests that the function f(x) is below the x-axis between x=2 and x=3. However, because probability cannot be negative, this inequality would not be true in the context of a probability density function.
Lastly, option (d), ∫3 to 4 f(x)dx ≠ 0, suggests that there is some non-zero area under the curve between x=3 and x=4. This could be true, unless f(x) is zero or there is equal area above and below the x-axis which would cancel out. Nonetheless, without specific information about f(x), we cannot state this with certainty.