Definite integrals-with modulus function The greatest value of the function $$f(x) =\int_1^x | t |dt$$ on the interval$[{-1\over2},{1\over2}]$is given by:
(1)$\frac 38$
(2)${-1\over2}$
(3)${-3\over 8}$
(4)${2\over 5}$ 
I have quite a problem dealing with this question.Should I take$f'(x)>0$?if yes then how should I proceed thereafter?

You can also solve this without any calculations:
Note that by the given interval the upper integration limit $x$ is always smaller than the lower limit $1$. For this reason, since the integrand is non-negative, the value of the integral is always negative. You can make it least negative (i.e. maximize it) by choosing the upper limit $x$ which differs least from the lower limit $1$. That's why you need to choose $x=1/2$ which gives $f(1/2)=-3/8$ as the answer.
