intervals

ryan1015

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Mar 1, 2010
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let g(x)=5x^4 +6x +3 a) find the intervals on which g is increasing and decreasing. b) find the intervals of concavity and the inflection points.


i know first derivative will show intervals of increase?decrease and 2nd derivative shows concavity and inflection points. my first derivative was 20x^3+6 and my 2nd derivative was 60x. i believe i set the first derivative to zero to get intervals of decrease and the same for 2nd derivative but i dont really know
 
What is ti that you are not knowing? You have given a complete description. Why not just finish up and be done? No need to doubt.

g(x) = 0 gives solutions or the equation or roots of g(x).

g'(x) = 0 gives potential min or max. Solve 20x^3 + 6 = 0 to find such points. The slope is positive or negative on each side. It may or may not change at the point.

g"(x) = 0 gives potential points of inflection. The concavity is positive or negative on each side. It may or may not change at the point. A point oi inflection will supersede a potential min or max.

Okay, now do it.
 
g(x) = 5x4+6x+3\displaystyle g(x) \ = \ 5x^{4}+6x+3

g(x) = 20x3+6 = 0, x =˙ .67\displaystyle g'(x) \ = \ 20x^{3}+6 \ = \ 0, \ x \ \dot= \ -.67

g(.67) =˙ .012\displaystyle g(-.67) \ \dot= \ -.012

g"(x) = 60x2, g"(.67) > 0, rel. min.\displaystyle g"(x) \ = \ 60x^{2}, \ g"(-.67) \ > \ 0, \ rel. \ min.

g"(x) = 60x2 = 0, x = 0\displaystyle g"(x) \ = \ 60x^{2} \ = \ 0, \ x \ = \ 0

Hence, (0,3) possible point of inflection, g"(1) > 0 and g"(1) > 0, ergo, no inflection point.\displaystyle Hence, \ (0,3) \ possible \ point \ of \ inflection, \ g"(-1) \ > \ 0 \ and \ g"(1) \ > \ 0, \ ergo, \ no \ inflection \ point.

Therefore, summing up, decreasing on (,.67] and increasing on [.67,), has a absolute\displaystyle Therefore, \ summing \ up, \ decreasing \ on \ (-\infty,-.67] \ and \ increasing \ on \ [-.67,\infty), \ has \ a \ absolute

 min. at (.67,.012), and no points of inflection, and is concave up throughout.\displaystyle \ min. \ at \ (-.67,-.012), \ and \ no \ points \ of \ inflection, \ and \ is \ concave \ up \ throughout.

See plot.\displaystyle See \ plot.

[attachment=0:27gpwrku]def.jpg[/attachment:27gpwrku]
 

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