Let's start by interpreting the information given in the problem:
1. Critical Point at [tex]\( x = 3 \)[/tex]: A critical point of a function is a point where the first derivative is zero or undefined. Since [tex]\( x = 3 \)[/tex] is a critical point of [tex]\( f(x) \)[/tex], we know that:
[tex]\[
f'(3) = 0
\][/tex]
2. Second Derivative at [tex]\( x = 3 \)[/tex]: The second derivative [tex]\( f''(3) = 2 \)[/tex] informs us about the concavity of the function at this point. Since [tex]\( f''(3) > 0 \)[/tex], the function has a local minimum at [tex]\( x = 3 \)[/tex].
Despite knowing this, we can’t determine [tex]\( f'(4) \)[/tex] simply from the given data. Here's why:
- The value of the first derivative at one point ( [tex]\( f'(3) \)[/tex] ) and the second derivative at that same point ( [tex]\( f''(3) \)[/tex] ) do not provide enough information to determine the first derivative at another point ( [tex]\( f'(4) \)[/tex] ). To know [tex]\( f'(4) \)[/tex], we would need more specific information about the functional form of [tex]\( f(x) \)[/tex].
Thus, the value of [tex]\( f'(4) \)[/tex] cannot be directly determined from the given data. We can state that:
[tex]\[
f'(3) = 0
\][/tex]
and
[tex]\[
f'(4) \text{ cannot be determined from the given information}
\][/tex]
To summarize:
- [tex]\( f'(3) \)[/tex] is 0 because [tex]\( x = 3 \)[/tex] is a critical point.
- [tex]\( f'(4) \)[/tex] cannot be determined based on the given information about the second derivative or the critical point at [tex]\( x = 3 \)[/tex].
Therefore, we conclude:
[tex]\[
f'(3) = 0, \quad \text{and} \quad f'(4) \text{ cannot be determined from the given information.}
\][/tex]
