What are the points of inflection, if any, of #f(x) =-3x^3 - 7x^2 + 3x#?

1 Answer
Narad T.
Jun 5, 2017

The point of inflection is #=(-0.778,-5.156)##

Explanation:

We calculate the first and second derivatives

#f(x)=-3x^3-7x^2+3x#

#f'(x)=-9x^2-14x+3#

#f''(x)=-18x-14#

The point of inflection is when

#f''(x)=0#

#-18x-14=0#, #=>#, #x=-14/18=-7/9#

Therefore, the point of inflection is

#(-7/9, f(-7/9))=(-0.778,-5.156)#

We can build a chart

#color(white)(aaaa)##x##color(white)(aaaa)##(-oo, -7/9)##color(white)(aaaa)##(-7/9,+oo)#

#color(white)(aaaa)##f''(x)##color(white)(aaaaaa)##+##color(white)(aaaaaaaaaaaaa)##-#

#color(white)(aaaa)##f(x)##color(white)(aaaaaaaa)##uu##color(white)(aaaaaaaaaaaaa)##nn#
graph{(y-(-3x^3-7x^2+3x))((x+0.778)^2+(y+5.156)^2-0.01)=0 [-11.24, 6.54, -8.084, 0.805]}

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