How do you find the critical numbers for #f(x) = 3x^4 + 4x^3 - 12x^2 + 5# to determine the maximum and minimum?
Critical points occur at
Critical point occur where the derivative of the function is equal to zero.
which can be factored as
Which implies the critical points (when
While not explicitly asked for in this question,
you can determine if each of these critical points is a minimum or maximum by evaluating the second derivative at each critical value.
Results greater than zero indicate a local minimum;
results less than zero indicate a local maximum;
results equal to zero indicate an inflection point.
Here is the graph for verification: