Use derivatives to find the critical points and inflection points of
\[
f(x)=x^{5}-10 x^{3}-12
\]
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 ：\(f'(x) = 5x^4 - 30x^2\)

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 ：\(\boxed{\text{Critical points: } x = 0, x = -\sqrt{6}, x = \sqrt{6}}\)

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