Answer:
The sign of f in the interval is positive.
Explanation:
1/2 < x > 5/3 is definitely wrong, because 1/2 is not greater than 5/3. The correct thing should be 1/2 < x < 5/3.
In order to know the sign of f on the interval 1/2 < x < 5/3 we choose a comfortable number in the interval and test in all the factors of f, then multiply the resulting signs from the factors.
The factors of f are:
2x – 1, 3x – 5, x – 4, 3x + 6. A comfortable number in the interval is 1. So let's test 1
2x – 1 = 2(1) – 1 = 2 – 1 = 1 is +ve
3x – 5 = 3(1) – 5 = 3 – 5 = –2 is -ve
x – 4 = 1 – 4 = –3 is -ve
3x + 6 = 3(1) + 6 = 3 + 6 = 9 is +ve. Now multiply the signs:
(+ve)(-ve)(-ve)(+ve) = +ve.
Therefore the sign of f in the interval is positive.