A beam of length $L$ and constant flexural rigidity $El$ is subjected to an applied moment $M_0$ at the end $x=L$ :

Total Points: $20$

The resulting deflection $y(x)$ is given by the following equation:

$y(x)=\frac{M_0}{6\text{E}\text{I}\text{L}}(L^{2}x-x^3).$

$($a$)$ Determine the location of the maximum deflection. $($Hint: find $(x)=d\frac{y}{d}x$; let $(x)=0$ and solve for $x)(10$ points$)$

$($b$)$ Determine the magnitude of the maximum deflection. $(5$ points$)$

$($c$)$ Determine the slope $(x)=d\frac{y}{d}x$ at $x=0.(5$ points$)$