Evaluate each of the following. Show all your calculations.
f" ����(−3�) if f(x) ���� = 4x^3�� − 3��x^2 + 2x� − 6
To evaluate the second derivative of the function f(x) = 4x^3 − 3x^2 + 2x − 6 at x = -3, we need to find the derivative of the derivative of f(x) with respect to x.
Step 1: Find the first derivative of f(x)
f'(x) = d/dx (4x^3 − 3x^2 + 2x − 6)
To find the derivative of each term, apply the power rule:
f'(x) = 12x^2 − 6x + 2
Step 2: Find the second derivative of f(x)
f''(x) = d/dx (12x^2 − 6x + 2)
Again, apply the power rule to differentiate each term:
f''(x) = 24x - 6
Step 3: Evaluate f''(-3)
Plug in x = -3 into the second derivative equation:
f''(-3) = 24(-3) - 6
f''(-3) = -72 - 6
f''(-3) = -78
Therefore, the value of the second derivative of f(x) at x = -3 is -78.