Question

Find the intervals of concavity and the inflection points of f(x)=2 x³+3 x⁴}-36 x .

Answer 1

To find the intervals of concavity and the inflection points of the given function, we find the second derivative and identify where it equals zero. We then test intervals to determine the concavity. The intervals of concavity are (-infinity, -1/3) and (0, infinity), and the points of inflection are x = -1/3 and x = 0.

Step-by-step explanation:

To find the intervals of concavity and the inflection points of the function, we need to analyze the second derivative of the function and solve for the points where the second derivative is equal to zero.

First, let's find the second derivative of the function.

Next, we set the second derivative equal to zero and solve for x to find the points of inflection:

After finding the critical points, we can determine the intervals of concavity by testing points in each interval.

Now, let's test the intervals:

So, the intervals of concavity are (-infinity, -1/3) and (0, infinity), and the points of inflection are x = -1/3 and x = 0.