Evaluate each of the following. Show all your calculations.
��� f" (1/3) �if f(x)���� = −2x^5�� + 2x� − 6
F(1/3) = -2*(1/3)^5 + 2*1/3 - 6
F(1/3) = -2/243 + 2/3 - 6
F(1/3)=-2/43 + 162/243 - 1458/243=
-1298/243 = -5.34.
To evaluate f"(1/3) for the given function f(x) = -2x^5 + 2x - 6, we need to find the second derivative of the function and then substitute x = 1/3 into the resulting expression.
First, let's find the first derivative of f(x):
f'(x) = d/dx (-2x^5 + 2x - 6)
= -10x^4 + 2
Next, let's find the second derivative by taking the derivative of f'(x):
f"(x) = d/dx (-10x^4 + 2)
= -40x^3
Now that we have the second derivative, we can substitute x = 1/3:
f"(1/3) = -40(1/3)^3
To simplify the expression, let's simplify the cube of (1/3):
(1/3)^3 = (1^3)/(3^3) = 1/27
Now, substitute this value back into the expression:
f"(1/3) = -40(1/27)
Finally, let's simplify further:
f"(1/3) = -40/27
Therefore, the value of f"(1/3) for the given function is -40/27.