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.