Determine The Left And Right Behavior Of The Graph Of The Polynomial Function f(x)=3+x-4x^4

f(x) = 3+x-4x^4.

To determine
the behaviour of the graph.

When x = 0 , y = 3. So the
graph makes an intercept of 3 on y axis.

When  y= 0, 
3+x-4x^3  is satisfied for x= 1.

Therefore x= 1, is an x
axis intercept.

Similarly there is one more intercept of x
axis between 0 and -1.

Also f'(x) = 0, gives = (3+x-4x^4)'
= 0.  So 1-16x^3 = 0 . Or x = (1/16)^(1/3) . So  f((1/16)^1/3) is an extreme
value.

f"(x) = (1-16x^3)' =
-48x^2.

f"((1/16)^(1/3) = -48(1/16)^(2/3) is
negative.

Therefore  f(1/16)^(1/3)) is the maximum at x= 
(1/16)^(1/3).

Also f(x)  is increasing  for  f'(x) =
1-16x^3 > 0  when x < (1/16)^(1/3).

f(x) is
decreasing  for 1-16x^3 < 0  when x >
(1/16)^(1/3).

The curve is extending downward idefinitely
in 4th and 3rd quadrants as x--> +infinity or x--> -
infinity.