Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 24

Sketch the graph of a function that all of the given conditons.
$f(0) = f'(0) = f'(2) = f'(4) = f'(6) = 0$,
$f'(x) > 0$ if $ 0 < x < 2$ or $4 < x < 6$,
$f'(x) < 0$ if $ 2 < x < 4$ or $x > 6$
$f''(x) > 0$ if $0 < x < 1$ or $3 < x < 5$
$f''(x) < 0$ if $1 < x < 3$ or $x > 5$, $f(-x) = f(x)$

The first statement tells us that these are horizontal tangets at $x = 0$, 2, 4 and 6 since the slope is equal to 0. Next condition tells us that the function is increasing at intervals $0 < x < 2$ and $4 < x < 6$. The third statement tells us that the function is decreasing at intervals $2 < x < 4$ and $x > 6$. The fourth statement tells us that there is an upward concavity at intervals $0 < x < 1$ and $3 < x < 5$. While the last statement tells us that there is a downward concavity at intervals $1 < x < 3$ and $x > 5$
So, the graph might look like this...