a) The function will be monotonic in any interval in which the derivative does not change sign. Find the derivative, see where it is zero, and use the location of the zero(s) to identify the desired intervals.
f'(x) = e^x -1
f'(0) = 0
The function is monotonically decreasing for x < 0 and monotonically increasing for x > 0.
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b) true.