How do you find the maximum, minimum and inflection points and concavity for the function #f(x)=6x^6+12x+6#?

1 Answer
Jun 20, 2016

Relative minimum at #x=(-1/3)^(1/5)#.
Always concave up and no inflection points.

Explanation:

#f'(x)=36x^5+12#
#f''(x) =180#

To find relative maxima, set #f'(x)=0#.
So, #36x^5+12=0#
#therefore# #x^5=-1/3#
#therefore x = (-1/3)^(1/5)#

Now, use the sign test to determine the relative maximum and minimum of the function.

For - infinity# < x < (-1/3)^(1/5)#, the function is decreasing.
For #(-1/3)^(1/5) < x< # infinity, the function is increasing.
Therefore, the function has relative minimum at #x = (-1/3)^(1/5)#

Regarding concavity of the function, since the second derivative is always positive, we can determine that the function is always concave up and has no inflection points.