Single Variable Calculus, Chapter 4, 4.4, Section 4.4, Problem 52

Sketch the graph of a function that satisfies the conditions

$f'(2) = 0, \quad f'(0) = 1, \quad f'(x) > 0 \quad \text{if } 0 < x < 2$,


$
\begin{array}{cc}
f'(x) < 0 & \text{ if } x > 2, \\
f''(x) < 0 & \text{ if } 0 < x < 4, \\
f''(x) > 0 & \text{ if } x > 4, \\
f''(x) > 0 & \text{ if } x > 4, \\
\lim_{x \to \infty} f(x) = 0, & \\
f (-x ) = - f(x) \text{ for all } x &
\end{array}
$


Based from the given conditions, it is an odd function that has horizontal asymptote at $y = 0$. There is a local maximum at $x = 2$. The inflection points are at $x = 0$ and $x = 4$. The function is concave down on $(0, 4)$ and concave up on $(4, \infty)$

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