(A) (6 marks) Let $f , g : [ 0,1 ] \mapsto \mathbb { R }$ be monotonically increasing continuous functions. Show that $$\left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right) \leq \int _ { 0 } ^ { 1 } f ( x ) g ( x ) d x$$ (Hint: try double integrals.) (B) (4 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R }$ be an infinitely differentiable function such that $f ( 1 ) = f ( 0 ) = 0$. Also, suppose that for some $n > 0$, the first $n$ derivatives of $f$ vanish at zero. Then prove that for the $( n + 1 )$ th derivative of $f$, $f ^ { ( n + 1 ) } ( x ) = 0$ for some $x \in ( 0,1 )$.
(A) (6 marks) Let $f , g : [ 0,1 ] \mapsto \mathbb { R }$ be monotonically increasing continuous functions. Show that
$$\left( \int _ { 0 } ^ { 1 } f ( x ) d x \right) \left( \int _ { 0 } ^ { 1 } g ( x ) d x \right) \leq \int _ { 0 } ^ { 1 } f ( x ) g ( x ) d x$$
(Hint: try double integrals.)\\
(B) (4 marks) Let $f : \mathbb { R } \longrightarrow \mathbb { R }$ be an infinitely differentiable function such that $f ( 1 ) = f ( 0 ) = 0$. Also, suppose that for some $n > 0$, the first $n$ derivatives of $f$ vanish at zero. Then prove that for the $( n + 1 )$ th derivative of $f$, $f ^ { ( n + 1 ) } ( x ) = 0$ for some $x \in ( 0,1 )$.