This page explores an alternative definition of concavity.
According to our text a differentiable function $f$ is, by definition, concave up on open interval $I$ if $f'$ is increasing on $I$. Consider the following alternative definition. Function $f$ is concave up on $(a,b)$ if for every $[c,d]\subset (a,b)$ the chord connecting $(c,f(c))$ to $(d,f(d))$ stays on or above the graph of $f$ over $[c,d]$. Is this definition equivalent to the one in the text? Why or why not? (Of course we can easily write an analogous alternative definition for concave down.)
The Sage interact below allows you to choose function $f$ and interval $(a,b)$ by text entry, then explore the relationship between the graph of $f$ on $(a,b)$ and chords on this graph by manipulating variable chord endpoints with a range slider.
Some suggested settings to explore:
$f(x)$: x^2 + 2*cos(2*x) $(a,b)$: (-1,1)
$f(x)$: 2*x + abs(x-1) $(a,b)$: (0,2)
$f(x)$: exp(x) + x + abs(x) $(a,b)$: (-1,1)