Evaluate each of the following. Show all your calculations.
f" ����(−3�) if f(x) ���� = 4x^3�� − 3��x^2 + 2x� − 6
Same procedure as a previous post.
2-4(2-7)
To evaluate the second derivative of the function f(x) at x = -3, we need to follow a step-by-step process.
Step 1: Find the first derivative of the function f(x).
Given f(x) = 4x^3 - 3x^2 + 2x - 6, we will differentiate each term using the power rule.
The power rule states that the derivative of xn is nx^(n-1).
Differentiating the first term, 4x^3:
The derivative of 4x^3 is 3 * 4 * x^(3-1) = 12x^2
Differentiating the second term, -3x^2:
The derivative of -3x^2 is 2 * -3 * x^(2-1) = -6x
Differentiating the third term, 2x:
The derivative of 2x is 1 * 2 * x^(1-1) = 2
Since -6 is a constant, its derivative is zero.
So, the first derivative of f(x) is:
f'(x) = 12x^2 - 6x + 2
Step 2: Find the second derivative of f(x).
Now, we will differentiate the first derivative we found in Step 1.
Differentiating the first term, 12x^2:
The derivative of 12x^2 is 2 * 12 * x^(2-1) = 24x
Differentiating the second term, -6x:
The derivative of -6x is -6
Since 2 is a constant, its derivative is zero.
So, the second derivative of f(x) is:
f''(x) = 24x - 6
Step 3: Substitute x = -3 into f''(x).
To evaluate f''(x) at x = -3, we substitute -3 for x in the equation:
f''(-3) = 24(-3) - 6
Calculating the expression:
f''(-3) = -72 - 6
Simplifying further:
f''(-3) = -78
Therefore, the value of the second derivative of f(x) at x = -3 is -78.