20. Given the function $f ( x ) = 4 x - x ^ { 4 } , x \in \mathbb{R}$. (1) Find the monotonicity of $f ( x )$; (2) Let $P$ be the intersection point of the curve $y = f ( x )$ and the positive $x$-axis. The tangent line to the curve at point $P$ is $y = g ( x )$. Prove that for any positive real number $x$, we have $f ( x ) \leq g ( x )$; (3) If the equation $f ( x ) = a$ (where $a$ is a real number) has two positive real roots $x _ { 1 } , x _ { 2 }$ with $x _ { 1 } < x _ { 2 }$, prove that $x _ { 2 } - x _ { 1 } < - \frac { a } { 3 } + 4 ^ { \frac { 1 } { 3 } }$.
20. Given the function $f ( x ) = 4 x - x ^ { 4 } , x \in \mathbb{R}$.\\
(1) Find the monotonicity of $f ( x )$;\\
(2) Let $P$ be the intersection point of the curve $y = f ( x )$ and the positive $x$-axis. The tangent line to the curve at point $P$ is $y = g ( x )$. Prove that for any positive real number $x$, we have $f ( x ) \leq g ( x )$;\\
(3) If the equation $f ( x ) = a$ (where $a$ is a real number) has two positive real roots $x _ { 1 } , x _ { 2 }$ with $x _ { 1 } < x _ { 2 }$, prove that $x _ { 2 } - x _ { 1 } < - \frac { a } { 3 } + 4 ^ { \frac { 1 } { 3 } }$.