Question

For what values of x is `f(x)= 7x^3 + 2 x^2 + 7x -2` concave or convex?

Answer 1

Concave down for `x < -2/21`, concave up for `x > -2/21`

Explanation:

First, we can try to find inflection points for this function. An inflection point is a point where the concavity changes, so finding this point is often helpful when analyzing concavity.

The process is straightforward; we will find `x` such that `d^2/dx^2 f(x) = 0`, that is, when the second derivative of `f` is zero.

Start by finding the 1st derivative, by simply applying the power rule to each term:

`d/dx f(x) = 21x^2 + 4x + 7`

Then, differentiate again to find the 2nd derivative:

`d^2/dx^2 f(x) = 42x + 4`

So, now we set the thing equal to zero:

`0 = 42x + 4`

And solve for `x`:

`x = -2/21`

So now we know that the function has one inflection point. What about the rest of possible values for `x`?

Well, if a segment of a graph is concave up (its slope is increasing) then the 2nd derivative will be positive. And if a segment is concave down, with a decreasing slope, the 2nd derivative will be negative.

Since we know that the 2nd derivative switches from negative to positive or vice versa at `x = -2/21`, let's see whether it's positive or negative, at, for instance, `x = 0`:

`d^2/dx^2 f(0) = 42*0 + 4 = 4`

Interesting. So the 2nd derivative is positive at `x=0`. This tells us that, since the only switch occurs at `x = -2/21`, all `x > -2/21` have a positive 2nd derivative as well, and are therefore concave up.

On the other hand, all `x < -2/21` are concave down, since the switch must occur at `x=-2/21`.

Hopefully this makes sense.