The function 3x^(3)+6x^(2)+6x+10 is maxima, minima or point of inflection?
1 Answer
 No mins or maxes
 Point of Inflection at
#x = 2/3# .
graph{3x^3 + 6x^2 + 6x + 10 [10, 10, 10, 20]}
Explanation:
Mins and Maxes
For a given
These values of
Note: Not all critical points are max/mins, but all max/mins are are critical points
So, let's find these for your function:
This doesn't factor, so let's try quadratic formula:
...and we can stop right there. As you can see, we end up having a negative number under the square root. Hence, there are no real critical points for this function.

Points of Inflection
Now, let's find points of inflection. These are points where the graphÂ has a change in concavity (or curvature). For a point (call it
Note: Not all such points are points of inflection, but all points of inflection must satisfy this.
So let's find these:
Now, we need to check if this is in fact a point of inflection. So we'll need to verify that
So let's test values to the right & left of
Right:
Left:
We don't care as much what the actual values are, but as we can clearly see, there's a positive number to the right of
To summarize,
Let's take a look at the graph of
graph{3x^3 + 6x^2 + 6x + 10 [10, 10, 10, 20]}
This graph is increasing everywhere, so it doesn't have any place where the derivative = 0. However, it does go from curved down (concave down) to curved up (concave up) at
Hope that helped :)