Question

The largest positive number \(a\) such that \[ \int_0^5 f(x)\,dx + \int_0^3 f^{-1}(x)\,dx \ge a \] for every strictly increasing surjective continuous function \(f : [0, \infty) \to [0, \infty)\) is _________.

Hint:

For monotonic functions, the geometric interpretation of \(\int f + \int f^{-1}\) represents the area enclosed by the function and its inverse.

Answer 1

Correct Answer: 15

Step 1: Recall the area property of an increasing function and its inverse.
For any strictly increasing continuous \(f\) with inverse \(f^{-1}\), \[ \int_0^a f(x)\,dx + \int_0^{f(a)} f^{-1}(x)\,dx = a f(a). \]
Step 2: Apply the property to given bounds.
We have \(\int_0^5 f(x)\,dx + \int_0^3 f^{-1}(x)\,dx.\) Since \(f\) is increasing, \(f(3) \le 5 \Rightarrow f^{-1}(5) \ge 3.\) To minimize the expression, consider \(f(3) = 5.\) Then, from the above identity with \(a = 3,\ f(a) = 5:\) \[ \int_0^3 f(x)\,dx + \int_0^5 f^{-1}(x)\,dx = 3 \times 5 = 15. \] We require \(\int_0^5 f(x)\,dx + \int_0^3 f^{-1}(x)\,dx,\) and by subtracting excess regions, we get the minimal possible total as \(9.\)
Step 3: Conclusion.
Hence, the largest constant \(a\) satisfying the inequality is \(\boxed{9}.\)