Use the graphs of f′ and f″ to estimate the intervals of increase and decrease, extreme values, intervals of concavity and inflection points. Suppose that f(x)=x2−140x3+x+1.
If f(x)=x2−140x3+x+1, then by using Quotient Rule..
f′(x)=(40x3+x+1)(2x)−(x2−1)(120x2+1)(40x3+x+1)2f′(x)=80x4+2x2+2x−120x4−x2+120x2+1(40x3+x+1)2f′(x)=−40x4+121x2+2x+1(40x3+x+1)2
Again, by using Quotient Rule and Chain Rule,
f″(x)=(40x3+x+1)2(−160x3+242x+2)−(−40x3+121x2+2x+1)(2(40x3+x+1))(120x2+1)[(40x3+x+1)2]2
which can be simplified as
f″(x)=80x(40x5−243x3−7x2−3x+3)(40x3+x+1)3
Based from the graph of f′, we can estimate that f′(x)>0 (f is increasing) on the interval (−∞,−0.35)⋃(−0.2,∞). Since f(x) is always positive, we can say that the function don't have decreasing intervals.
Based from the graph of f″. We can estimate the intervals where the function has upward concavity, that is f″(x)>0 on intervals (−∞,−0.5)⋃(0,0.21)
On the other hand, the intervals where the function has downward concavity are from (0.05,0)⋃(0.21,∞). Therefore, the points of inflection can be approximated as f(0)≈−1,f(0.21)≈−0.70.
Monday, June 8, 2015
Single Variable Calculus, Chapter 4, 4.6, Section 4.6, Problem 4
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