How do you find points of inflection and determine the intervals of concavity given #y=4x^3+21x^2+36x-20#?
There is an inflection point at
Concavity and inflection points of a function can be determined by looking at the second derivative. If the second derivative is 0, it is an inflection point (IE where the graph changes concavity). If the second derivative is positive, you know the graph is concave up (
So let's start by finding the second derivative for our function,
Now we can solve the following equation to find the inflection point(s):
Now, let's look at the intervals before and after our only inflection point.
If we pick a point smaller than
If we then pick a point larger than
In conclusion, by analyzing the function's second derivative we figured out that it has an inflection point at