Finding the Range of an Integrated Function with Given Constraints

[WubbaLubba Dubdub]

Homework Statement

Let ##g(x) = \int_0^xf(t) dt## where ##f## is such that ##\frac{1}{2} \leq f(t) \leq 1## for ##t \in [0,1]## and ##\frac{1}{2} \geq f(t) \geq 0## for ##t \in (1,2]##. Then ##g(2)## belongs to interval
A. ##[\frac{-3}{2}, \frac{1}{2}]##
B. ##[0, 2)##
C. ##(\frac{3}{2}, \frac{5}{2}]##
D. ##(2, 4)##

Homework Equations

The Attempt at a Solution

I got ##g'(x) = f(x)## and using this and the definite integral given, i have ##g(0) = 0##
I didn't really know where to go from here, so I tried making a graph (sort of) using the minimum and maximum slopes of the function in the given intervals and found an area in which, I think the function will exist, with the interval for ##g(2)## being ##[\frac{1}{2},\frac{3}{2}]##.

This isn't present in the options...Can someone please point out my mistakes and help me get the answer.

 

[BvU]

I mostly agree with your reasoning; generally the lower bound of an interval is given first, followed by the upper bound: ##[{1\over 2},{3\over 2}]## is your result. If it isn't in the list litterally, you might try to exclude the answers that certainly don't satisfy...

 

[WubbaLubba Dubdub]

I mostly agree with your reasoning; generally the lower bound of an interval is given first, followed by the upper bound: ##[{1\over 2},{3\over 2}]## is your result. If it isn't in the list litterally, you might try to exclude the answers that certainly don't satisfy...

Ah silly me. Will edit it. I found the answer key too and it says B is correct. Is ##g(0) =0## correct?

 

[BvU]

Ah silly me. Will edit it.
better leave as is or the thread becomes unintellegible...

I found the answer key too and it says B is correct.
yes. If it belongs to [1/2, 3/2] it certainly belongs to [0,2). The others all miss something

Is ##g(0) =0## correct?
yes.

 

[WubbaLubba Dubdub]

Now I get it! Thank you!