Question

The function 3x^(3)+6x^(2)+6x+10 is maxima, minima or point of inflection?

Answer 1

• 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 `x`-value (let's call it `c`) to be a max or min for a given function, it has to satisfy the following:

`f'(c) = 0` or undefined.

These values of `c` are also called your critical points.

Note: Not all critical points are max/mins, but all max/mins are are critical points

So, let's find these for your function:

`f'(x) = 0`

`=> d/dx(3x^3 + 6x^2 + 6x + 10) = 0`

`=> 9x^2 + 12x + 6 = 0`

This doesn't factor, so let's try quadratic formula:

`x = (-12 +- sqrt(12^2 - 4(9)(6)))/(2(9))`

`=> (-12 +-sqrt(-72))/18`

...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 graph has a change in concavity (or curvature). For a point (call it `c`) to be a point of inflection, it must satisfy the following:

`f''(c) = 0`.

Note: Not all such points are points of inflection, but all points of inflection must satisfy this.

So let's find these:

`f''(x) = 0`

`=> d/dx(d/dx(3x^3 + 6x^2 +6x + 10)) = 0`

`=> d/dx(9x^2 + 12x + 6 = 0)`

`=> 18x + 12 = 0`

`=> x = -12/18 = -2/3`

Now, we need to check if this is in fact a point of inflection. So we'll need to verify that `f''(x)` does in fact switch sign at `x = -2/3`.

So let's test values to the right & left of `x = -2/3`:

Right:
`x = 0`
`f''(0) = 12`

Left:
`x = -1`
`f''(-1) = -6`

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 `x = -2/3`, and a negative number to the left of `x = -2/3`. Hence, it is indeed a point of inflection.

To summarize, `f(x)` has no critical points (or mins or maxes), but it does have a point of inflection at `x = -2/3`.

Let's take a look at the graph of `f(x)` and see what these results mean:

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 `x = -2/3`.

Hope that helped :)