Question

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

Answer 1

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.