Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 22

Sketch the graph of a function that satisfies all of the given conditions
$f'(1) = f'(-1) = 0$, $f'(x) < 0$ if $|x| < 1$,
$f'(x) > 0$ if $ 1 < |x| < 2$, $f'(x) = -1$ if $|x| > 2$,
$f''(x) < 0$ if $-2 < x < 0$, inflection point (0,1)




The first statement tells us that there is an horizontal tangents at $x = 1$ and $ x = -1$ since the slope is zero. The second statement tells us that the slope is positive at interval. $1 < |x| < 2$ and slope of -1 at interval $|x| > 2$. The last statement tells us that the inflection point should be at (0,1) and should have a downward concavity at interval $ -2 < x < 0$