1. Suppose that f is an integrable function and that Find Find Find Explain why f(x) must be
negative somewhere in the interval [1, 2].
Solution
a) 04 f(x)dx =02 f(x)dx +24 f(x)dx = 1 + 7 = 8
so, 04 f(x)dx = 8
b) 10 f(x)dx = -01 f(x)dx = - (2)
so,10 f(x)dx = -2
c) as, 01 f(x)dx + 12 f(x)dx = 02 f(x)dx,
so, 2 + 12 f(x)dx = 1
so,12 f(x)dx = 1 - 2
or,12 f(x)dx = -1
d) as we saw in (c), the integration value is negative in interval [1,2]
so either some portion of function f(x) will be below Y-axis in [1.2] or even the whole function
is always negative in [1,2]