Question

Assume f is continuous on [a, b] satisfying f(x) > 0. Prove that

∫[a, b] lim (n→[infinity]) (1/n) Σ[1 to n] f(xi) dxi = max f(xi) for all xi in [a, b].

Answer 1

The question is based on a common misunderstanding in integral calculus regarding the limit of Riemann sums and continuous functions. While the Riemann sum approaches the definite integral over an interval, it does not equate to the function's maximum value.

Step-by-step explanation:

The student has posed a problem related to integral calculus, specifically concerning the limit of a Riemann sum and its equivalence to the maximum value of a continuous function over a closed interval.

However, the statement presented by the student appears incorrect, as it suggests the limit of the sum of these function values over the interval equals the maximum function value which is not generally true.

For a continuous function f that is positive on an interval [a, b], as the number of partitions n goes to infinity, the Riemann sum approaches the definite integral of f from a to b, not necessarily the maximum value of f.

If we consider a continuous probability density function, the area under the curve between two points on the x-axis represents the probability that a value in that interval is observed.

By the definition of a probability density function, the integral of f over its entire domain equals 1, if it expresses a total probability, and not specifically the maximum value of f.