Final answer:
To find all values of 'x' for which p'(x) = 0, we differentiate p(x) to get p'(x) = 3x² - 28x + 9 and then set this derivative equal to zero, solving the quadratic equation that results to find the values of x.
Step-by-step explanation:
The student is asked to compute the derivative of the polynomial p(x) = x³ - 14x² + 9x + 58, and then use this derivative, denoted as p'(x), to find all values of x for which p'(x) = 0. The first step is differentiation, which gives us the derivative:
p'(x) = 3x² - 28x + 9.
To find the values of x for which p'(x) = 0, we set the derivative equal to zero:
3x² - 28x + 9 = 0
This is a quadratic equation and can be solved using the quadratic formula or factoring if possible. The solutions to this equation will give us the values of x at which the slope of the original function p(x) is zero. These values correspond to potential maximum and minimum points on the graph of p(x).