Chapter 4 - Applications of the Derivative - 4.2 What Derivatives Tell Us - 4.2 Exercises - Page 259: 104

If f is concave up at $x = c$, then $f'(c) > 0$. This means that $f$ is increasing in the neighborhood near $c$. So for $x < c$, the slope of the function should be less than the slope of the tangent line at $c$, and for $x > c$ it should be greater than the slope of the tangent line. This means that the curve is “bending upward”, away from itself, and this indicates that the tangent line should be below the curve in a neighborhood near $c$.