How are $f^{\prime}(0)$ and $M$ related?

$f(x)$ defined in all reals satisfies the following:

1. $f(x)$ is differentiable.

2. $f(x)-f(2-x)=0, f(x)+f(-x)=0$

3. $f(x)=-\dfrac1{f^{\prime}(0)}x$ has only three roots, that is $0,α,-α$ .

Let the maximum value of $f(x)$ be $M$ .

QUESTION Is there an explicit equation between $f^{\prime}(0)$ and $M$ ?

#Calculus

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

How are f′(0)f^{\prime}(0)f′(0) and MMM related?

Comments

How are $f^{\prime}(0)$ and $M$ related?