Question - Determining Interval for Local Maximum of a Function
Solution:
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)
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