How do we tell if the graph of a function is concave upward? The second derivative is greater than 0 The second derivative is 0 The second derivative is smaller than 0 The second derivative is undefined
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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 $h(x)=$ $f(x)-f\left(x_{0}\right)-f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right) .$ Show that $h$ has a local minimum value at $x_{0}$ and hence that $h(x) \geq 0$ on the interval. Show that $h(x)>0$ if $x \neq x_{0}$.
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