For what values of x is #f(x)= 7x^3 + 2 x^2 + 7x -2 # concave or convex?
Concave down for
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
Start by finding the 1st derivative, by simply applying the power rule to each term:
Then, differentiate again to find the 2nd derivative:
So, now we set the thing equal to zero:
And solve for
So now we know that the function has one inflection point. What about the rest of possible values for
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
Interesting. So the 2nd derivative is positive at
On the other hand, all
Hopefully this makes sense.