Consider the function f(x)=1−x^3, for 0≤x≤1. Describe a partition P of [0,1] into 4 intervals of equal length and find the largest and smallest values of f(x) on each of them.

Question

Consider the function f(x)=1−x³, for 0≤x≤1. Describe a partition P of [0,1] into 4 intervals of equal length and find the largest and smallest values of f(x) on each of them.

Answer 1

I will answer this question in two ways. For the first way, I will assume that you are allowed the use of a graphing calculator. For the second way, I will assume that you are not allowed the use of a graphing calculator. As the partition applies to both ways, I will explain that now. What you are being asked to do is look at only the inclusive interval of 0 through 1, then divide that interval into 4 equal parts. To do this, divide the length of the whole interval (which is 1 in this case) by 4. 1/4 = 0.25, so each partition of the interval has a length of 0.25. This is along the same axis as the now-divided interval, which is the x-axis in this case. You are then asked to find the largest and smallest y-values on each partitioned section. First method: Graph the equation on your calculator, then input for a minimum bounded by the first section (x=0 and x=0.25 in this case), then the maximum of the same section, then repeat for the other 3 sections.

Second way: Calculate all mins and maxes for the entire curve. In this case, there are only two: negative infinity and infinity, respectively. This means that the mins and maxes of a partitioned section must be located at the bounds of that partitioned section. Now all you have to do is plug those x values into the equation of the curve and look at which one is higher for each of the four partitioned sections.