Final answer:
To find the critical points of the function, take the derivative of the function and set it equal to zero. Solve the resulting equation to find the values of x that make the derivative zero. These values are the critical points.
Step-by-step explanation:
To find the critical points of the function f(x) = 2x³ + 15x² + 36x, we need to find the values of x where the derivative of the function equals zero. First, we find the derivative of f(x) by applying the power rule: f'(x) = 6x² + 30x + 36. Next, we set the derivative equal to zero and solve for x:
6x² + 30x + 36 = 0
This quadratic equation can be factored as:
2(x² + 5x + 6) = 0
Then, we solve for x by setting each factor equal to zero:
x² + 5x + 6 = 0
(x + 2)(x + 3) = 0
Setting each factor equal to zero, we find x = -2 and x = -3. Therefore, the critical points of the function are x = -2 and x = -3.