Determining Interval for Local Maximum of a Function
The question asks us to find on which interval the function f is guaranteed to attain a local maximum, based on the selected values of its first derivative f'(x) given in the table.
A local maximum occurs where the derivative changes from positive to negative. Looking at the table of values for f'(x), we see:
- f'(0) = -1, meaning the derivative is negative just before x=0.
- f'(1) = 2, meaning the derivative is positive at x=1.
- f'(2) = -3, meaning the derivative is negative at x=2.
- f'(3) = 0, meaning the derivative is zero at x=3.
- f'(4) = 5, meaning the derivative is positive at x=4.
From these values, we notice that f'(x) goes from positive at x=1 to negative at x=2. This indicates that there is a local maximum somewhere between x=1 and x=2, as per the first derivative test which says that if the derivative changes from positive to negative at some point, then this point is a local maximum.
Therefore, the correct interval on which f is guaranteed to attain a local maximum is:
(B) (1,2)
