Let $f(x)=x^2$ if $x0$ and $f(x)=-x^2$ if Is $0$ a critical point of $f?$ Does $f$ have an inflection point there? Is $f^{\text{'}\text{'}}(0)=0?$ If a function has a nonvertical tangent line at an inflection point, does the second derivative of the function necessarily vanish at that point?

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Verify that if $f$ is concave up on an interval, then its graph lies above its tangent lines on that interval. Hint: Suppose $f$ is concave up on an open interval containing $x_0.$ Let $h(x)=f(x)-f(x_0)-f^{\text{'}}(x_0)(x-x_0).$ Show that $h$ has a local minimum value at $x_0$ and hence that $h(x)0$ on the interval. Show that $h(x)>0$ if $x{x}_0.$