How do you find the inflection points of the graph of the function: #y=x^3-15x^2+33x+100#?

1 Answer
Bill K.
Jul 22, 2015

The inflection point is #(x,y)=(5,15)#.

Explanation:

If #f(x)=x^3-15x^2+33x+100#, then the first derivative is #f'(x)=3x^2-30x+33# and the second derivative is #f''(x)=6x-30=6(x-5)#. The second derivative changes sign (from negative to positive) as #x# increases through #x=5#, making this the first coordinate of the unique inflection point. In this case, the inflection point occurs where the graph of #y=f(x)# changes from concave down (frown-like) to concave up (smile-like).

The second coordinate of the inflection point is #f(5)=5^3-15*5^2+33*5+100#

#=125-375+165+100=15#.

The inflection point is therefore #(x,y)=(5,f(5))=(5,15)#

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