Short Answer 
The analysis begins by determining that ( b = 24 ) using the fact that f(-24) = 0. Then, it concludes that ( a < 0 ) because f(24) must be negative. Among the evaluated statements, only ( D: a < b ) is correct.
Step 1: Determine the Value of b
We start with the function f(x) = a‚Äöao(x + b) and know that the point (-24, 0) is on its graph, which means f(-24) = 0. By substituting -24 into the function, we find:
- f(-24) = a‚Äöao(-24 + b) = 0
- Since a ‚a† 0, we conclude that ‚ao(-24 + b) = 0, leading to b = 24.
Step 2: Determine the Value of a
Next, we know that f(24) < 0. Plugging 24 into the function gives us:
- f(24) = a‚Äöao(24 + b) = a‚Äöao(48)
- Since ‚Äöao(48) is positive, for f(24) to be negative, a must be negative.
- Thus, we can conclude that a < 0.
Step 3: Evaluate the Statements
Finally, we evaluate the options which include claims about the values of a and b. Since we know b = 24 and a < 0, we check each statement:
- A: f(0) = 24 – This is incorrect as a‚Äöao(24) cannot equal 24.
- B: f(0) = -24 – This is also incorrect as it does not guarantee a specific outcome.
- C: a > b – Incorrect, as a is negative while b is positive.
- D: a < b – Correct, since a is negative and b = 24.
Thus, the only true statement is (D) a < b.