Final answer:
No, ƒ(x) does not have to be equal to 3 for all x. The fact that ƒ′(x) = 0 for all x indicates that ƒ(x) is a constant function, but the constant can be any value, not just 3.
Step-by-step explanation:
No, ƒ(x) does not have to be equal to 3 for all x. We know that ƒ(-1) = 3, but we are also given that ƒ′(x) = 0 for all x. This means that the derivative of ƒ(x) with respect to x is always 0, indicating that ƒ(x) is a constant function. However, the constant could be any value, not just 3.
For example, if we have ƒ(x) = 2, then ƒ(-1) = 2 and ƒ(x) = 2 for all x.
Therefore, ƒ(x) does not have to be equal to 3 for all x, as long as ƒ′(x) = 0 for all x.