Let $f(x)$ be a function which is continuous for all $x$. Let $L_{100}, R_{100}$ and $M_{100}$ be the Riemann sums using 100 subintervals with left, right and middle sample points, respectively, for $f$ on the interval $[0,10]$. Which of the following statements is FALSE? $L_{100} \leq M_{100} \leq R_{100}$ If $f$ is decreasing on $[0,10]$, then $L_{100} \geq R_{100}$ If $f$ is increasing on $[0,10]$, then $L_{100} \leq R_{100}$. All three sums $L_{100}, R_{100}, M_{100}$ exist. The definite integral $\int_{0}^{10} f(x) d x$ exists (i.e. $f$ is Riemann integrable on $[0,10]$ ).
Solution
Step 1 ：The problem is asking which of the given statements about Riemann sums and definite integrals is false.
Step 2 ：The first statement is generally true for a function that is increasing on the interval [0,10]. The left Riemann sum (L100) will be an underestimate, the right Riemann sum (R100) will be an overestimate, and the middle Riemann sum (M100) will be somewhere in between.
Step 3 ：The second statement is generally true for a function that is decreasing on the interval [0,10]. The left Riemann sum (L100) will be an overestimate, and the right Riemann sum (R100) will be an underestimate.
Step 4 ：The third statement is generally true for a function that is increasing on the interval [0,10]. The left Riemann sum (L100) will be an underestimate, and the right Riemann sum (R100) will be an overestimate.
Step 5 ：The fourth statement is true because the function is continuous on the interval [0,10], and therefore, the Riemann sums exist.
Step 6 ：The fifth statement is true because the function is continuous on the interval [0,10], and therefore, the function is Riemann integrable on this interval.
Step 7 ：Therefore, the first statement is false because it assumes that the function is increasing on the interval [0,10]. If the function is decreasing, then the left Riemann sum (L100) will be an overestimate, the right Riemann sum (R100) will be an underestimate, and the middle Riemann sum (M100) will be somewhere in between.
Step 8 ：Final Answer: The false statement is \(L_{100} \leq M_{100} \leq R_{100}\).
