Question: First, prove that if a function \(f:[a, b] \rightarrow R\) is continuous and a function \(g:[a, b] \rightarrow R\) is integrable and nonnegative, then there exists \(c \in[a, b]\) such that \(\int_{a}^{b} f(x) g(x) d x=f(c) \int_{a}^{b} g(x) d x\) Second, use this result to prove that \(f(x)=f(a)+\sum_{k=1}^{n} \frac{f^{(k)}(a)}{k !}(x-a)^{k}+\frac{1}{n !} \int_{a}^{x} f^{(n+1)}(t)(x-t)^{n} d t\),

where \(f^{(n+1)}(t)\) is continuous.

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