Sure, let's go through the process of finding the second derivative of the function [tex]\( f(x) = 4x^4 - 3x^3 - 3x^2 + 7 \)[/tex]. Then, we will evaluate the second derivative at [tex]\( x = 0 \)[/tex] and [tex]\( x = 8 \)[/tex].
1. First Derivative:
We start by finding the first derivative [tex]\( f'(x) \)[/tex] of the function [tex]\( f(x) \)[/tex].
[tex]\[
f(x) = 4x^4 - 3x^3 - 3x^2 + 7
\][/tex]
To find [tex]\( f'(x) \)[/tex], we differentiate [tex]\( f(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
f'(x) = \frac{d}{dx}(4x^4 - 3x^3 - 3x^2 + 7)
\][/tex]
Using the power rule [tex]\( \left(\frac{d}{dx} x^n = nx^{n-1}\right) \)[/tex], we get:
[tex]\[
f'(x) = 16x^3 - 9x^2 - 6x
\][/tex]
2. Second Derivative:
Next, we need to find the second derivative [tex]\( f''(x) \)[/tex] by differentiating [tex]\( f'(x) \)[/tex]:
[tex]\[
f'(x) = 16x^3 - 9x^2 - 6x
\][/tex]
Differentiate [tex]\( f'(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
f''(x) = \frac{d}{dx}(16x^3 - 9x^2 - 6x)
\][/tex]
Again, applying the power rule:
[tex]\[
f''(x) = 48x^2 - 18x - 6
\][/tex]
Therefore, the second derivative of the function [tex]\( f(x) \)[/tex] is:
[tex]\[
f''(x) = 48x^2 - 18x - 6
\][/tex]
3. Evaluating [tex]\( f''(0) \)[/tex]:
To find [tex]\( f''(0) \)[/tex], we substitute [tex]\( x = 0 \)[/tex] into [tex]\( f''(x) \)[/tex]:
[tex]\[
f''(0) = 48(0)^2 - 18(0) - 6 = -6
\][/tex]
4. Evaluating [tex]\( f''(8) \)[/tex]:
To find [tex]\( f''(8) \)[/tex], we substitute [tex]\( x = 8 \)[/tex] into [tex]\( f''(x) \)[/tex]:
[tex]\[
f''(8) = 48(8)^2 - 18(8) - 6
\][/tex]
Calculate each term separately:
[tex]\[
48(8)^2 = 48 \times 64 = 3072
\][/tex]
[tex]\[
18(8) = 144
\][/tex]
Substite these values into [tex]\( f''(8) \)[/tex]:
[tex]\[
f''(8) = 3072 - 144 - 6 = 2922
\][/tex]
So, the second derivative is:
[tex]\[
f''(x) = 48x^2 - 18x - 6
\][/tex]
And the specific evaluations are:
[tex]\[
f''(0) = -6
\][/tex]
[tex]\[
f''(8) = 2922
\][/tex]
