OUR SOLUTION OF PROBLEM: 2

Find the interval in which 
f(x)=2log(x-2)-xsq+4x+1 increases
a) (1,2)
b) (2,3)
c) (5/2,3)
d) (2,4)

Note: xsq term stands for x square.                         (DC/M)

SOLUTION:

Before starting up the problem, let’s first of all find out the domain of the given function

Since, we have (x-2) is inside the log it should be positive, it will happen only when x>2.

Hence the domain of given function is x > 2 therefore option a) is ruled out.

Now differentiate the given function, we get f ‘(x)= -(x-3)(x-1)/ (x-2)

For f(x) to increasing, f ‘ (x) > 0

Since, x> 2, the denominator of f ‘ (x ) is always positive, which means for f (x) to increasing -(x-3)(x-1) >0

which means x will lie from ( 1,3), but since the function is defined for x >2, we will have (2,3 )as the interval where function is increasing.

Hence the correct choices are (2,3) and (2.5 ,3)