Ecan read from these graphs that the absolute minimum is about 0 incorrect: your answer is incorrect. and occurs when x ≈ 1.5 incorrect: your answer is incorrect. in the second graph. there is also a local maximum ≈ 0.00002 when x ≈ in the third graph and a local minimum ≈ 211 when x ≈ in the last graph. these graphs also show three inflection points near −35, −5, and −1 and two between −1 and 0. to estimate the inflection points closely we would need to graph f '', but to compute f '' by hand is an unreasonable chore. if you have a computer algebra system, then it's easy to do. (see this exercise.)

Farmers know that driving heavy equipment on wet soil compresses the soil and injures future crops. here are data on the "penetrability" of the same type of soil at two levels of compression. penetrability is a measure of how much resistance plant roots will meet when they try to grow through the soil. compressed soil 2.85 2.66 3 2.82 2.76 2.81 2.78 3.08 2.94 2.86 3.08 2.82 2.78 2.98 3.00 2.78 2.96 2.90 3.18 3.16 intermediate soil 3.17 3.37 3.1 3.40 3.38 3.14 3.18 3.26 2.96 3.02 3.54 3.36 3.18 3.12 3.86 2.92 3.46 3.44 3.62 4.26 use the data, omitting the high outlier, to give a 95% confidence interval for the decrease in penetrability of compressed soil relative to intermediate soil. compute degrees of freedom using the conservative method. interval: to