Determine the critical points for each of the following functions, and determine whether the function has a local maximum value, a local minimum value, or neither at the critical points. Sketch the graph of each function.

a$.f(x)=-2x^2+8x+13$

d$.f(x)=-3x^3-5x$

b$.f(x)=\frac{1}{3}{x}^3-9x+2$

e$.f(x)=\sqrt[\phantom{2}]{x^2-2x+2}$

c$.f(x)=2x^3+9x^2+12x$

f$.f(x)=3x^4-4x^3$