Final answer:
To find the concavity of the function f(x) = 2/x^2 - 3/x^3, we can take its second derivative and evaluate it at the given point x = 3. By finding the second derivative and plugging in the value, we get -28/243, which is negative. Therefore, the function is concave downward at x = 3.
Step-by-step explanation:
A function is concave upward when its second derivative is positive, and concave downward when its second derivative is negative. To find the concavity of the function f(x) = 2/x^2 - 3/x^3, we need to take the second derivative. Let's start by finding the first derivative:
f'(x) = -4/x^3 + 9/x^4
To find the second derivative, we differentiate f'(x):
f''(x) = 12/x^4 - 36/x^5
Now, we can plug in the value x = 3 into the second derivative:
f''(3) = 12/3^4 - 36/3^5 = 12/81 - 36/243 = 4/27 - 4/9 = -28/243
Since the second derivative f''(3) is negative, the function f(x) = 2/x^2 - 3/x^3 is concave downward at x = 3.