91Ó°ÊÓ

For each of the following statements, determine whether it is true or false and justify your answer. a. If \(f:[a, b] \rightarrow \mathbb{R}\) is integrable and \(\int_{a}^{b} f=0,\) then \(f(x)=0\) for all \(x\) in \([a, b]\) b. If \(f:[a, b] \rightarrow \mathbb{R}\) is integrable, then \(f:[a, b] \rightarrow \mathbb{R}\) is continuous. c. If \(f:[a, b] \rightarrow \mathbb{R}\) is integrable and \(f(x) \geq 0\) for all \(x\) in \([a, b],\) then \(\int_{a}^{b} f \geq 0\) d. A continuous function \(f:(a, b) \rightarrow \mathbb{R}\) defined on an open interval \((a, b)\) is bounded. e. A continuous function \(f:[a, b] \rightarrow \mathbb{R}\) defined on a closed interval \([a, b]\) is bounded.

(The Cauchy-Schwarz Inequality for Integrals) Suppose that the functions \(f, g, f^{2}\) \(g^{2},\) and \(f g\) are integrable on the closed, bounded interval \([a, b] .\) Prove that $$ \int_{a}^{b} f g \leq \sqrt{\int_{a}^{b} f^{2}} \sqrt{\int_{a}^{b} g^{2}} $$. (Hint: For each number \(\lambda,\) define \(p(\lambda)=\int_{a}^{b}[f-\lambda g]^{2}\). Show that \(p(\lambda)\) is a quadratic polynomial for which \(p(\lambda) \geq 0\) for all \(\lambda\) and therefore that its discriminant is not positive.)

Suppose that the bounded function \(f:[a, b] \rightarrow \mathbb{R}\) has the property that for each rational number \(x\) in the interval \([a, b], f(x)=0 .\) Prove that $$ \int_{a}^{b} f \leq 0 \leq \int_{a}^{b} f $$

Suppose the continuous function \(f:[a, b] \rightarrow \mathbb{R}\) has the property that $$ \int_{c}^{d} f \leq 0 \quad \text { whenever } a \leq c

The monotonicity property of the integral implies that if the functions \(g:[0, \infty) \rightarrow\) \(\mathbb{R}\) and \(h:[0, \infty) \rightarrow \mathbb{R}\) are continuous and \(g(x) \leq h(x)\) for all \(x \geq 0,\) then $$\int_{0}^{x} g \leq \int_{0}^{x} h \quad \text { for all } x \geq 0$$ Use this and the First Fundamental Theorem (Integrating Derivatives) to show that each of the following inequalities implies its successor: $$\begin{aligned} \cos x & \leq 1 & & \text { if } x \geq 0 \\ \sin x & \leq x & & \text { if } x \geq 0 \\\ 1-\cos x & \leq \frac{x^{2}}{2} & & \text { if } x \geq 0 \\ x-\sin x & \leq \frac{x^{3}}{6} & & \text { if } x \geq 0 . \end{aligned}$$ Thus, \(x-\frac{x^{3}}{6} \leq \sin x \leq x \quad\) if \(x \geq 0\)

Suppose that \(f:[a, b] \rightarrow \mathbb{R}\) is a bounded function for which there is a partition \(P\) of \([a, b]\) with \(L(f, P)=U(f, P) .\) Prove that \(f:[a, b] \rightarrow \mathbb{R}\) is constant.

(The Triangle Inequality for Integrals) Suppose that the functions \(f:[a, b] \rightarrow \mathbb{R}\) and \(g:[a, b] \rightarrow \mathbb{R}\) are continuous. Prove that $$ \int_{a}^{b}|f+g| \leq \int_{a}^{b}|f|+\int_{a}^{b}|g|. $$

Suppose that the functions \(f:[a, b] \rightarrow \mathbb{R}\) and \(g:[a, b] \rightarrow \mathbb{R}\) are integrable. Show that there is a sequence \(\left\\{P_{n}\right\\}\) of partitions of \([a, b]\) that is an Archimedean sequence of partitions for \(f\) on \([a, b]\) and also an Archimedean sequence of partitions for \(g\) on \([a, b] .\) (Hint: Use the Refinement Lemma.)