Determine which of the following is true for the function X^3+5x^2-8x+3

f(x) has a relative minimum at x = 2/3.
f(x) has a relative maximum at x = 2/3
f(x) has a zero at x = 2/3
Both A and C are true. <-----
Both B and C are true.

dy/dx = 3 x^2 + 10 x -8 yes 0

d^2y/dx^2 = 6 x + 10 positive so min

at 0
I get about .2 so not 0

so I think only A

To determine which statements are true for the function f(x) = x^3 + 5x^2 - 8x + 3, you can analyze the function's critical points and use the first derivative test.

1. Relative minimum at x = 2/3:
To find the critical points, you need to solve for f'(x) = 0. Take the derivative of f(x) with respect to x:

f'(x) = 3x^2 + 10x - 8.

To find the critical points, set f'(x) = 0 and solve the equation:
3x^2 + 10x - 8 = 0.

Using the quadratic formula or factoring, you can find that x = -2 and x = 2/3 are the solutions.

To determine whether x = 2/3 is a relative minimum, you need to test the values of f'(x) on either side of x = 2/3. Calculate f'(x) for x values smaller and larger than 2/3.

For x = 1, f'(1) = 3(1)^2 + 10(1) - 8 = 5.
For x = 1/2, f'(1/2) = 3(1/2)^2 + 10(1/2) - 8 = 4.

Since f'(1) > 0 and f'(1/2) > 0, it means that the function is increasing on both sides of x = 2/3. Therefore, x = 2/3 is not a relative minimum.

2. Relative maximum at x = 2/3:
Using the same reasoning, you can see that x = 2/3 is not a relative maximum. So, statement B is not true.

3. Zero at x = 2/3:
To check if x = 2/3 is a zero of the function, you need to evaluate f(2/3).

f(2/3) = (2/3)^3 + 5(2/3)^2 - 8(2/3) + 3 = 8/27 + 20/9 - 16/3 + 3 = 0.

Since f(2/3) = 0, it means that x = 2/3 is a zero of the function.

Therefore, statement C is true.

In conclusion, the correct option is "Both A and C are true."