For the given function
-----------------------(1)
we have to see when it will increase and decrease.
For that we will first find derivative of function
Now derivative of first term in f(x) which is -4 will be 0 as its constant.
then derivative of x is 1, so derivative of -x will be -1
To derivate last term of function,
, we will use power rule formula:

so

constant 3 will come as it is, so derivative of 3x^2 wil be 3(2x)= 6x.
So f'(x) = 0 -1 +6x
f'(x) = -1 + 6x------------------------------(2)
For function to be increasing f'(x) should be positive and for function to be decreasing f'(x) shoould be negative
So we will first find where f'(x) = 0
So put 0 in f'(x) place in equation (2)
0 = -1 + 6x
Now solve for x as shown
0 +1 = -1 + 6x +1
1 = 6x


So now we can have two regions as shown on either side of 1/6 on number line
_____________I______________
_______II__________
So to test region I, pick any number to the left side of
. For example lets take 0. now plug 0 inx place in f'(x) equation given by (2)
f'(x)= -1 + 6x
f'(0) = -1 + 6(0) = -1+0 = -1 which is negative. So since f'(x) we got as negative for region I, so this will be decreasing.
Interval for region I will be (-∞,
)
----------------------------------------------------------------------------------------------------------
similarly now test region II. For that pick any number to the right of
, lets take 1. So plug 1 in x place in f'(x) equation given by (2)
f'(x) = -1 + 6x
f'(1) = -1 +6(1)= -1+6 = 5 which is positive so we will have f(x) increasing in this region.
Interval for region II will be (
, ∞)
------------------------------------------------------------------------
Graphs of
and
are shown in attachment. So you can clearly see in graph that f'(x) is negative (below x axis) from -∞ till
so f(x) should be decreasing in this part which we can see from f(x) graph that its decreasing from -∞ till
.
Similarly f'(x) is positive( above x axis) beyond
so its increasing beyond
which we can veryify from f(x) graph we can see that its increasing from
till ∞. Hence verified