Use derivatives to find the critical points and inflection points of
$$f(x)=x^5-10x^3-12$$
Find all critical and inflection points.

Step 1 ：First, find the first and second derivatives of the function $f(x)=x^5-10x^3-12$:

Step 2 ：$f^{\prime }(x)=5x^4-30x^2$

Step 3 ：$f^{″}(x)=20x^3-60x$

Step 4 ：Find the critical points by setting the first derivative equal to zero and solving for x:

Step 5 ：$5x^4-30x^2=0$

Step 6 ：Critical points: $x=0,x=-\sqrt{6},x=\sqrt{6}$

Step 7 ：Find the inflection points by setting the second derivative equal to zero and solving for x:

Step 8 ：$20x^3-60x=0$

Step 9 ：Inflection points: $x=0,x=-\sqrt{3},x=\sqrt{3}$

Step 10 ：$\boxed{{\displaystyle \text{Critical points: }x=0,x=-\sqrt{6},x=\sqrt{6}}}$

Step 11 ：$\boxed{{\displaystyle \text{Inflection points: }x=0,x=-\sqrt{3},x=\sqrt{3}}}$