91Ó°ÊÓ

(a) Given the function \(f: x \rightarrow x^{3}\) defined on \(I=\\{x: 0 \leqslant x \leqslant 1\\} .\) Suppose \(\Delta\) is a subdivision and \(\Delta^{\prime}\) is a refinement of \(\Delta\) which adds one more point. Show that $$ S^{+}\left(f, \Delta^{\prime}\right)S_{-}(f, \Delta) $$ (b) Give an example of a function \(f\) defined on \(I\) such that $$ S^{+}\left(f, \Delta^{\prime}\right)=S^{+}(f, \Delta) \quad \text { and } \quad S_{-}\left(f, \Delta^{\prime}\right)=S_{-}(f, \Delta) $$ for the two subdivisions in Part (a). (c) If \(f\) is a strictly increasing continuous function on \(I\) show that \(S^{+}\left(f, \Delta^{\prime}\right)<\) \(S^{+}(f, \Delta)\) where \(\Delta^{\prime}\) is any refinement of \(\Delta\)

Prove that \(e^{x} \geqslant 1+x\) for all \(x\).

Prove the following theorem: Suppose \(f\) and \(g\) are integrable on \([a, b]\) and \(f(x) \leqslant\) \(g(x)\) on \([a, b]\). Suppose also that \(S=\\{(x, y): a \leqslant x \leqslant b, f(x) \leqslant y \leqslant g(x)\\}\). Then \(S\) is a figure and $$ A(S)=\int_{a}^{b}[g(x)-f(x)] d x $$

Suppose that \(f\) is continuous on \(I=\\{x: a \leqslant x \leqslant b\\}\) except at an interior point \(c\). If \(f\) is also bounded on \(I\), prove that \(f\) is integrable on \(I\). Show that the value of \(f\) at \(c\) does not affect the value of \(\int_{a}^{b} f(x) d x\). Conclude that the value of \(f\) at any finite number of points cannot affect the value of the integral of \(f\).

Suppose that \(f\) is continuous, nonnegative and not identically zero on \(I=\) \(\\{x: a \leqslant x \leqslant b\\}\). Prove that \(\int_{a}^{b} f(x) d x>0\). Is the result true if \(f\) is not continuous but only integrable on \(I ?\)

Suppose that \(f\) is continuous on \(I=\\{x: a \leqslant x \leqslant b\\}\) and that \(\int_{a}^{b} f(x) d x=0\). If \(f\) is nonnegative on \(I\) show that \(f \equiv 0\).