$\lambda$ for which the function $f(x)=2x^3-3(2+\lambda)x^2+12\lambda x$ has exactly one local maxima and exactly one local minima

You found critical points correctly.

Take the second derivative: $$f''=12x-12-6\lambda.$$

Two cases:

$1) f''(2)>0, \ f''(\lambda)<0 \Rightarrow \lambda<2$ or

$2) f''(2)<0, \ f''(\lambda)>0 \Rightarrow \lambda>2.$

Hence: $\lambda\ne 2.$

Answer: all but $B$.

No need for considering the second derivative: as $f'(x)=6x^2-6(\lambda+2)x+12\lambda$, the condition is $f'(x)$ to have two critical values, and this happens if and only if the quadratic has a (reduced) discriminant $$\Delta'=9(\lambda+2)^2-72\lambda=9\bigl((\lambda+2)^2-8\lambda\bigr)=9(\lambda-2)^2>0.$$ This condition is satisfied exactly when $\lambda\ne2$.

HINT:

The derivative of a cubic will be a quadratic.

A quadratic can have (at best) two distinct roots.

How does one decide on the number of roots a quadratic has?

If your cubic has a one local max, and one local min then it has two distinct turning points.