The critical numbers of this function are 3, 1.

Critical numbers are values of c at which, for the function `f(c)` , `f'(c) = 0` or where `f'(c)` is undefined.

To solve the function `f(x) = x^3 - 6x^2 + 9x + 7`, you first need to find the derivative of f(x). Since this is a polynomial, you can simply use the power rule. To use the power rule, multiply the coefficient of each variable by the variable's exponent, subtract 1 from the original exponent, and simplify. For constants, the derivative is equal to 0.

This gives us `f '(x) = 3 x ^ 2 -12x +9`.

After doing this, you can find the critical numbers by setting `f'(x)` to zero and solving. This gives us the numbers 3 and 1.