Final answer:
To find where the function f(x) = 2x^4 + 16x^3 + 36x^2 is both decreasing and concave down, calculate the first derivative (f'(x)) to find decreasing intervals and the second derivative (f''(x)) for concavity. Solve the inequalities for f'(x) < 0 and f''(x) < 0 and find their intersection.
Step-by-step explanation:
To determine the intervals on which the function f(x) = 2x^4 + 16x^3 + 36x^2 is both decreasing and concave down, we need to find the first and second derivatives of f(x).
The first derivative, f'(x), gives us the slope of f(x) at any point, which helps us determine where the function is increasing or decreasing. The intervals where f'(x) is less than zero indicate where f(x) is decreasing.
The second derivative, f''(x), tells us the concavity of the function. If f''(x) is less than zero, then the function is concave down on that interval.
First, we calculate f'(x):
f'(x) = 8x^3 + 48x^2 + 72x
Next, we find where f'(x) is less than zero to see where f(x) is decreasing.
Then, we calculate f''(x):
f''(x) = 24x^2 + 96x + 72
Lastly, we look for intervals where f''(x) is less than zero to determine where f(x) is concave down.
We need to find the intersection of these intervals to answer the question. This process involves solving inequalities and possibly using a sign chart or graphing technology.