Final answer:
To find the relative maximum of the function f(x), we differentiated it to find the critical points, solved the resulting quadratic equation, and determined that the relative maximum occurs at x = -1.
Step-by-step explanation:
To determine the relative maximum of the function f(x) = 3x5 - 4x3 - 3x, we need to find the critical points by taking the derivative and setting it equal to zero. The derivative of the function is f'(x) = 15x4 - 12x2 - 3. Setting the derivative equal to zero gives us the critical points:
f'(x) = 0
15x4 - 12x2 - 3 = 0
This is a quadratic equation in terms of x2, let's denote u = x2, so we have:
15u2 - 12u - 3 = 0
Solving this quadratic equation:
(5u + 1)(3u - 3) = 0
This gives us u = -1/5 or u = 1. Since u = x2 and x2 cannot be negative, we reject u = -1/5 and only consider u = 1. Therefore, x can be either +1 or -1. To determine which one is a relative maximum, we test these points by calculating the second derivative to check concavity or we can use a sign chart for the first derivative.
After evaluating, we find that the relative maximum occurs at x = -1. Thus, the answer is a) -1.