Mathematics

Mathematics

Problem

Which of the following sets of conditions is sufficient to guarantee a number c∈(a,b)c\in(a,b) satisfying ∫abg(x) dx=g(c)(b−a)\displaystyle \int_a^b g(x)\,dx=g(c)(b-a)? a) gg is integrable on [a,b][a,b] only. b) gg is differentiable on (a,b)(a,b) but may be discontinuous at aa or bb. c) gg is continuous on [a,b][a,b]. d) gg is bounded on [a,b][a,b] and has finitely many discontinuities.
Intuitive explanation

This problem is about the Mean Value Theorem for Integrals, which says a continuous function must equal its average value somewhere in the interval. The key insight is recognizing which conditions on gg force it to be continuous (or at least well-behaved enough) so that it actually attains its average value 1b−a∫abg(x) dx\frac{1}{b-a}\int_a^b g(x)\,dx at some point cc in (a,b)(a,b). Look for conditions that guarantee continuity or eliminate pathological behavior like jump discontinuities.

Intuitive explanation

This problem is about the Mean Value Theorem for Integrals, which says a continuous function must equal its average value somewhere in the interval. The key insight is recognizing which conditions on gg force it to be continuous (or at least well-behaved enough) so that it actually attains its average value 1b−a∫abg(x) dx\frac{1}{b-a}\int_a^b g(x)\,dx at some point cc in (a,b)(a,b). Look for conditions that guarantee continuity or eliminate pathological behavior like jump discontinuities.

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