Final answer:
The correct value of b for which f'(1) = 9, where f(x) = log_b(9x³ - 8), is found to be b = 3 after deriving and solving for b.
Step-by-step explanation:
The student asked for the value of b for which f'(1) = 9 where f(x) = log_b(9x³ - 8). To solve this, we need to calculate the derivative of the function and then substitute x = 1 to find the corresponding b.
First, recall that the derivative of log_b(u) with respect to x is (1 / (u ln(b))) * du/dx where u is a function of x. In this case, u = 9x³ - 8, so du/dx = 27x². At x = 1, u becomes 1 and du/dx becomes 27. Therefore, the derivative of f at x = 1 is (27 / (1 ln(b))).
To find b, we set 27 / (1 ln(b)) = 9 and solve for b. This simplifies to 3 = 1 / ln(b) which gives ln(b) = 1 / 3. Taking the exponential of both sides gives b as e^(1 / 3). We can then test the given options to see which one is closest to e^(1 / 3).
Upon evaluation, the closest value is when b = 3, which is e^(1 / 3) = 1.396... Therefore, option b) is the correct answer.