Spring work from $x_1$ to $x_2$: W = $\frac{1}{2}$k($x_1^2$ - $x_2^2$). The work by the external agent is the negative of this.

Critical trap: Work from 0 to x is \frac{1}{2}$$kx^2. Work from x to 2x is \frac{3}{2}$$kx^2 (NOT another \frac{1}{2}$$kx^2). The spring force increases with deformation, so more work is needed for the same additional stretch.

Maximum compression when all KE converts to PE: \frac{1}{2}$$mv^2 = \frac{1}{2}$$kx_{max}^2, giving $x_{max}$ = v*sqrt$\frac{m}{k}$.

For a body falling height h onto a spring: mg(h+x) = \frac{1}{2}$$kx^2 (must include the extra distance x the body falls during compression).