For what values of a and b does the function f(x)=ax^3+bx have a minimum or maximum at x=2?

dy/dx = 0 at max or min = 3 a x^2 + b

so when
3 a (4) + b = 0
b = -12 a

To determine the values of a and b for which the function f(x) = ax^3 + bx has a minimum or maximum at x = 2, we need to use calculus.

First, let's find the derivative of f(x) with respect to x. The derivative represents the rate at which the function is changing at a given point. For this function, we have:

f'(x) = 3ax^2 + b

To find the critical points, we equate the derivative to zero and solve for x:

3ax^2 + b = 0

Since we are specifically looking for x = 2, let's substitute x = 2 into the equation:

3a(2^2) + b = 0
12a + b = 0

This equation gives us a relationship between a and b. Now, let's determine whether this critical point is a maximum or a minimum.

To do so, we need to find the second derivative, which represents the concavity of the function. We take the derivative of f'(x):

f''(x) = 6ax

Again, substitute x = 2 into the equation:

f''(2) = 6a(2) = 12a

If f''(2) is positive, then it confirms that x = 2 is a minimum point. If it is negative, then x = 2 is a maximum point. So, let's consider two scenarios:

1. If f''(2) > 0:
12a > 0
This implies a > 0

2. If f''(2) < 0:
12a < 0
This implies a < 0

Therefore, if a > 0, the function f(x) = ax^3 + bx has a minimum at x = 2, and if a < 0, it has a maximum at x = 2. The value of b does not affect whether there is a maximum or minimum, only the value of a does.