Question

Use calculus to find the maximum and minimum of f(x)=x³−6x²+9x−1 on the interval [0,5).

a) Maximum at x=0, minimum at x=3
b) Maximum at x=3, minimum at x=0
c) Maximum at x=5, minimum at x=0
d) Maximum at x=5, minimum at x=3

Answer 1

To find the function's maximum and minimum, we differentiate and solve for critical points, then evaluate at endpoints of the interval. The maximum occurs at x = 3 and the minimum at x = 0.

Step-by-step explanation:

To find the maximum and minimum of the function f(x) = x³ - 6x² + 9x - 1 on the interval [0,5), we first need to find the critical points by taking the derivative of the function and setting it equal to zero. The derivative is f'(x) = 3x² - 12x + 9.

Setting the derivative equal to zero gives us the quadratic equation 3x² - 12x + 9 = 0, which simplifies to x² - 4x + 3 = 0. By factoring, we obtain (x - 1)(x - 3) = 0, giving us potential critical points at x = 1 and x = 3.

We also need to examine the endpoints of the interval, which are x = 0 and x = 5. However, since the interval is from [0,5), x = 5 is not included. By evaluating the function at the critical points and the endpoint, we find that at x = 0, f(x) = -1, at x = 1, f(x) = 3, and at x = 3, f(x) = 3. Therefore, the maximum value of the function on the given interval is at x = 3 and the minimum is at x = 0.