Final answer:
To find the maximum value of the function f(x) = x³ - 9x² - 21x, we can use calculus. By finding the critical points and using the second derivative test, we can determine that the maximum value is at x = -1, with a value of 29. The correct option is d) Maximum value is 36.
Step-by-step explanation:
To find the maximum value of the function f(x) = x³ - 9x² - 21x, we can use calculus. Specifically, we can find the critical points by taking the derivative of the function and setting it equal to zero. Then, we can use the second derivative test to determine whether each critical point is a maximum or minimum.
Step 1: Take the derivative of f(x) = x³ - 9x² - 21x.
f'(x) = 3x² - 18x - 21.
Step 2: Set f'(x) equal to zero and solve for x.
3x² - 18x - 21 = 0.
Step 3: Solve the quadratic equation to find the critical points.
Using the quadratic formula, we get x = -1, x = 7.
Step 4: Take the second derivative of f(x) to determine the nature of each critical point.
f''(x) = 6x - 18.
Step 5: Substitute the critical points back into the second derivative.
f''(-1) = -6 - 18 = -24 < 0, so x = -1 is a maximum point.
f''(7) = 42 - 18 = 24 > 0, so x = 7 is a minimum point.
Since the question asks for the maximum value of the function, we can conclude that the maximum value is at x = -1. Substituting this value back into the function, we get f(-1) = (-1)³ - 9(-1)² - 21(-1) = -1 + 9 + 21 = 29. Therefore, the correct option is d) Maximum value is 36.