Final answer:
To find a function f such that f'(x) = 3x³ and the line 81x + y = 0 is tangent to the graph of f, we need to find the function that satisfies these conditions. The function can be determined by finding the antiderivative of 3x³, which is (3/4)x⁴ + C, where C is any constant.
Step-by-step explanation:
To find a function f such that f'(x) = 3x³ and the line 81x + y = 0 is tangent to the graph of f, we need to find the function that satisfies these conditions. Since the line is tangent to the graph of f, the slope of the line is equal to the derivative of f at the point of tangency. In this case, the slope of the line is 81.
Therefore, the derivative of f at the point of tangency must also be 81. We can find the antiderivative of 3x³ to get the function f(x). The antiderivative of 3x³ is (3/4)x⁴ + C, where C is a constant. So, f(x) = (3/4)x⁴ + C, where C is any constant.