Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$ be a solution of the differential equation (1). For each real $t \geq 0$ we set, $T \left( x ^ { \prime } \right) ( t ) = \frac { 1 } { 2 } \left\langle A x ^ { \prime } ( t ) ; x ^ { \prime } ( t ) \right\rangle$ and $U ( x ) ( t ) = \frac { 1 } { 2 } \langle K x ( t ) ; x ( t ) \rangle$. Show then that the quantity $T \left( x ^ { \prime } \right) ( t ) + U ( x ) ( t )$ does not depend on $t \in [ 0 , + \infty [$.

Given below is the plot of a potential energy function $\mathrm { U } ( \mathrm { x } )$ for a system, in which a particle is in one dimensional motion, while a conservative force $\mathrm { F } ( \mathrm { x } )$ acts on it. Suppose that $\mathrm { E } _ { \text {mech} } = 8 \mathrm {~J}$, the incorrect statement for this system is:
[where K.E. = kinetic energy]
(1) at $\mathrm { x } > \mathrm { x } _ { 4 }$, K. E. is constant throughout the region.
(2) at $\mathrm { x } < \mathrm { x } _ { 1 }$, K. E. is smallest and the particle is moving at the slowest speed.
(3) at $\mathbf { x } = \mathbf { x } _ { 2 }$, K. E. is greatest and the particle is moving at the fastest speed.
(4) at $x = x _ { 3 }$, K.E. $= 4 \mathrm {~J}$