A statically indeterminate beam of length L and flexural rigidity EI is cantilevered at the left end $x=0,$ simply supported at the right end $x=L$ and subjected to a uniform distributed load w :

The equation of the deflection curve $y(x)$ from $0$ to $L$ is given by

$y(x)=\frac{w{a}^2}{6\text{E}\text{I}\text{L}}x(L^2-x^2)$

a$)$ Determine the equation for the slope $(x)=d\frac{y}{d}x.(4$ points$)$

b$)$ Evaluate both the deflection and slope at the points $x=0$ and $x=L.(4$ points$)$

c$)$ Determine both the location and magnitude of the maximum deflection. $(6$ points$)$

d$)$ Use your results of parts a$)-$c$)$ to sketch the deflection $y(x).$ Clearly label the axes and any $ma\frac{x}{m}in$ points along the beam. $(6$ points$).$

e$)$ Comment on your findings from resolving the deflection, slope and magnitude of the maximum deflections? Make it make sense to a non$-$engineering colleague. $(5$ Points$)$