Ranking Left, Right, and Midpoint sums by shape of the graph

Since $f$ is decreasing on $[2,10]$, each left endpoint gives a larger function value than the corresponding right endpoint on the same subinterval, so the Left sum overestimates the area and the Right sum underestimates it. This immediately tells us $R_4 < L_4$, since every rectangle used in $R_4$ is shorter than the corresponding rectangle in $L_4$.

Since $f$ is also concave up, we can apply the property that for a concave up function, the secant line connecting two points on the curve lies below the curve itself. On each subinterval, the midpoint approximation uses the function value at the midpoint, which lies above the secant line connecting the endpoints. Therefore, $M_4$ overestimates the true area. However, $M_4$ overestimates by less than $L_4$ does, because the left-endpoint sum uses the larger endpoint values at the left of each subinterval, while the midpoint uses intermediate values. Combining both observations gives the ranking $R_4 < M_4 < L_4$.