Discuss the following relation s about upper and lower Riemann sums

Question

Let , $f$ be a continuously differentiable real-valued function on $[a,b]$ such that $|f'(x)|\le k $ for all $x\in [a,b]$. For a partition $P=\{a=a_0<a_1<a_2<\cdots <a_n=b\}$ let $U(P,f)$ and $L(P,f)$ denotes the upper and lower Riemann sums of $f$ with respect to $P$. Then ,

(A) $|L(P,f)|\le k(b-a)\le|U(P,f)|$

(B) $U(P,f)-L(P,f)\le k(b-a)$

(C) $U(P,f)-L(P,f)\le k||P||$ , where $||P||=\max_{0\le i\le n-1}(a_{i+1}-a_i)$

(D) $U(P,f)-L(P,f)\le k||P||(b-a)$

I really can't understand how I utilize the condition $|f'(x)|\le k$ ?

Can anyone give me any hint ?

Answer 1

By the mean value theorem, for all $x,y \in [a,b]$, $$|f(x) - f(y)| \leq k(x-y)$$ Let $x_{M_i}$ and $x_{m_i}$ denote the $x$-coordinates of the maximum and minimum of the function in each subinterval. Then \begin{align*} U(P,f) - L(P,f) &= \sum_{i=0}^n (M_i - m_i) \Delta x_i \\ &= \sum_{i=0}^n \left(f(x_{M_i}) - f(x_{m_i})\right) \Delta x_i \\ &\leq \sum_{i=0}^n k(x_{M_i} - x_{m_i}) \Delta x_i \\ &\leq k ||P|| \sum_{i=0}^n \Delta x_i \\ &= k ||P|| (b-a). \end{align*}