Question

Consider the graph of f(x) = (x-4)^3
on which intervals is f concave up

Answer 1

First, let's find the first derivative of f(x):
f'(x) = 3 (x-4)^2
Now, let's find the second derivative of f(x):
f"" (x) = 6 (X-4)
To determine the intervals where f(x) is concave up, we need to find where f(x) > 0.
Setting f'(x) > 0:
6(X-4) > 0
X-4 > 0
× > 4
Therefore, f(x) is concave up for all x values greater than 4. In interval notation, this is written as (4, ∞).