How do I find concave up and concave down from f(x)=x^3+3x^2+5x+7?

1 Answer
Bill K.
May 22, 2015

First find the derivative: f'(x)=3x^2+6x+5. Next find the second derivative: f''(x)=6x+6=6(x+1). The second derivative changes sign from negative to positive as x increases through the value x=1.

Therefore the graph of f is concave down when x<1, concave up when x>1, and has an inflection point when x=1.

The coordinates of the inflection point are (1,f(1))=(1,16).

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