MathDB

(1) Let

f(x)

be a continuous function defined on

[a,\ b]

, it is known that there exists some

c

such that \int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b) Explain the fact by using graph. Note that you don't need to prove the statement. (2) Let f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n, Prove that there exists

\theta

such that f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.

Problem Statement