# Discuss the following relation s about upper and lower Riemann sums

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 $$P=\{a=a_0 let $$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 ?

• At first glance, maybe you should try to use the Mean Value Theorem. Aug 22, 2015 at 3:49

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*}