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QUESTION

What is the x-coordinate of the point of inflection on the graph of ##y=1/10x^(5)+1/2X^(4)-3/10##?

We find the Inflection Points of ##y## by finding the second derivative of the function (y''), and the x-values at which y'' equals 0.

We look for the zeroes because at those points the concavity (or the direction in which the slope of the function ##f(x)## is trending) has leveled off; it is at these points that the concavity is most likely to turn from positive to negative, or vice-versa.

Just as background, Math is Fun offers notes on inflection points.

##y= 1/10x^5 +1/2x^4 -3/10## ##y'=1/2x^4+ 2x^3## ##y''=2x^3+6x^2##

We set our second derivative to 0: ##y''=x^2(2x+6)=0##

And we can find our inflection points through these equations:

  • ##x^2=0##
  • ##2x+6=0##

Our Inflection Points are then at ##x= 0, and x=-3##

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