Specifically I want to interpret what f"(x)=0 means geometrically.
Geometric meaning of double derivatives
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geometry
derivatives
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1$f'' = 0$ on an interval $\implies f'(x)=b$ is constant $\implies f(x)=ax+b$ for $a,b \in \Bbb R$. Any convex and concave function is affine. – 2017-01-26
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0[These notes might be helpful](http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx) – 2017-01-26
1 Answers
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If $f''(x)=0$ for all $x$, that means $f(x)$ is a linear function.
On the other hand, if $f(x)$ is not linear, you set $f''(x)$ equal to $0$ (and solve for $x$) to find the points of inflection of $f(x)$. The points of inflection are where the graph of the function changes from concave up to concave down and vice versa.