Evaluate each of the following. Show all your calculations.
��� f" (1/3) �if f(x)���� = −2x^5�� + 2x� − 6
To evaluate f " (1/3) for the given function f(x) = -2x^5 + 2x - 6, we need to find the second derivative of f(x) and then substitute x = 1/3 into it.
Here's how to find the second derivative of f(x):
1. Find the derivative of f(x) with respect to x:
f'(x) = d/dx(-2x^5 + 2x - 6)
= -10x^4 + 2
2. Find the derivative of f'(x) with respect to x:
f "(x) = d/dx(-10x^4 + 2)
= -40x^3
Now, to evaluate f "(1/3), substitute x = 1/3 into f "(x):
f "(1/3) = -40(1/3)^3
= -40/27
Therefore, the value of f "(1/3) is -40/27.