Final answer:
To determine the critical numbers of the function, calculate its derivative, set it equal to zero, and solve for x.
Step-by-step explanation:
To identify the critical numbers of the function f(x) = x⁴ + 12x³ + 16x² + 8, we first need to find the function's derivative and set it equal to zero to solve for the points where the slope of the tangent to the curve is zero, highlighting the potential maxima, minima, or points of inflection.
The derivative of the function is: f'(x) = 4x³ + 36x² + 32x. Setting this derivative equal to zero gives us the equation 4x³ + 36x² + 32x = 0. Factoring out a common term x, we get x(4x² + 36x + 32) = 0. This can be solved by factoring further or using the quadratic formula if necessary, to find the critical numbers.