This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You're on a roll — let's determine the equation of the function f(x). Given the function f(x) = -x^3 + ax^2 + bx + c. Step 1: Use the information about the point of inflection. A point of inflection occurs where the second derivative f''(x) is equal to zero. First, find the first derivative f'(x): f'(x) = -3x^2 + 2ax + b Next, find the second derivative f''(x): f''(x) = -6x + 2a The point of inflection is at (0; 27), so f''(0) = 0. -6(0) + 2a = 0 2a = 0 a = 0 Also, the point (0; 27) lies on the function f(x), so f(0) = 27. f(0) = -(0)^3 + a(0)^2 + b(0) + c = 27 c = 27 Now we know a=0 and c=27. Substitute these values back into f(x): f(x) = -x^3 + (0)x^2 + bx + 27 f(x) = -x^3 + bx + 27 Step 2: Use the information f(-3) = 0. Substitute x=-3 into the updated f(x): f(-3) = -(-3)^3 + b(-3) + 27 = 0 -(-27) - 3b + 27 = 0 27 - 3b + 27 = 0 54 - 3b = 0 3b = 54 b = (54)/(3) b = 18 Step 3: Write the final equation of f(x). We have found a=0, b=18, and c=27. Substitute these values into the original form of f(x): f(x) = -x^3 + (0)x^2 + (18)x + 27 f(x) = -x^3 + 18x + 27 Drop the next question.