Question

Consider a non-negative function \( f(x) \) which is continuous and bounded over the interval [2, 8]. Let \( M \) and \( m \) denote, respectively, the maximum and the minimum values of \( f(x) \) over the interval. Among the combinations of \( \alpha \) and \( \beta \) given below, choose the one(s) for which the inequality \[ \beta \leq \int_2^8 f(x) \, dx \leq \alpha \] is guaranteed to hold.

• A. \( \beta = 5 \, m, \, \alpha = 7 \, M \)
• B. \( \beta = 6 \, m, \, \alpha = 5 \, M \)
• C. \( \beta = 7 \, m, \, \alpha = 6 \, M \)
• D. \( \beta = 7 \, m, \, \alpha = 5 \, M \)

Hint:

To estimate the area under a curve of a bounded function, use the minimum and maximum values of the function as lower and upper bounds, respectively. This gives an effective way to approximate the integral of the function.

Answer 1

The Correct Option is A

The inequality can be understood as describing the total area under the curve \( f(x) \) over the interval [2, 8]. Given that \( f(x) \) is bounded, the minimum value \( m \) and maximum value \( M \) represent the lower and upper bounds of the function. The total area under the curve (which is the integral) will lie between: \[ \int_2^8 f(x) \, dx \geq 5 \, m \cdot (8 - 2) = 30 \, m \] and \[ \int_2^8 f(x) \, dx \leq 7 \, M \cdot (8 - 2) = 42 \, M. \] Thus, the inequality \( \beta \leq \int_2^8 f(x) \, dx \leq \alpha \) will hold if \( \beta = 5 \, m \) and \( \alpha = 7 \, M \), so the correct answer is (A).