Question

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

Answer 1

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)`.