# Find the following for function #f(x)=\ln(x^4+1)#...?

##
- intervals of increase/decrease
- local min/max values
- intervals of concavity and inflection points

- intervals of increase/decrease
- local min/max values
- intervals of concavity and inflection points

##### 2 Answers

For

#### Explanation:

The graph below vividly answers all your questions. Yet, there is a

formal way of finding all.

So, for

and at

Also, by successive differentiation, it is seen that

also 0 at x = 0. But

derivative

At

inflexion.

So,

and for

Of course, at x = 0, it is flat. At

nor convex.

The tangent y = 0 does not cross the the curve here, at x = 0 but it

does at the points of inflexion

this edition .are attributed to the nice observation in the comment by

Jim I hope that the answer is now alright, in all aspects.

graph{y-ln(1+x^4)=0 [-10, 10, -5, 5]}

See below.

#### Explanation:

The denominator is always positive, so the sign of

So,

# = (12x^6+12x^2-16x^6)/(x^4+1)^2#

# = (-4x^6+12x^2)/(x^4+1)^2#

# = (-4x^2(x^4+3))/(x^4+1)^2#

On

On

On

On

The inflection changes at