To find the second derivative of the function \( f(x) = 3 e^{-x^4} \), we can refer to the given second derivative formula:
[tex]\[ f^{\prime \prime}(x) = 12 x^2 \left( 4 x^4 - 3 \right) e^{-x^4} \][/tex]
First, let's find \( f^{\prime \prime}(0) \).
### Finding \( f^{\prime \prime}(0) \):
Substitute \( x = 0 \) into the second derivative:
[tex]\[ f^{\prime \prime}(0) = 12 \cdot 0^2 \left( 4 \cdot 0^4 - 3 \right) e^{-0^4} \][/tex]
Since \( 0^2 = 0 \) and \( e^{-0} = 1 \):
[tex]\[ f^{\prime \prime}(0) = 12 \cdot 0 \cdot \left( -3 \right) \cdot 1 \][/tex]
[tex]\[ f^{\prime \prime}(0) = 0 \][/tex]
So, the correct choice is:
A. \( f^{\prime \prime}(0) = 0 \)
### Finding \( f^{\prime \prime}(1) \):
Next, substitute \( x = 1 \) into the second derivative:
[tex]\[ f^{\prime \prime}(1) = 12 \cdot 1^2 \left( 4 \cdot 1^4 - 3 \right) e^{-1^4} \][/tex]
Calculating inside the parentheses first:
[tex]\[ 4 \cdot 1^4 - 3 = 4 \cdot 1 - 3 = 4 - 3 = 1 \][/tex]
So, it reduces to:
[tex]\[ f^{\prime \prime}(1) = 12 \cdot 1 \cdot 1 \cdot e^{-1} \][/tex]
[tex]\[ f^{\prime \prime}(1) = 12 \cdot e^{-1} \][/tex]
Using the given value of \( f^{\prime \prime}(1) \):
[tex]\[ f^{\prime \prime}(1) = 4.4146 \][/tex]
Thus, the correct choice is:
A. \( f^{\prime \prime}(1) = 4.4146 \)
Therefore, the finalized answers are:
- \( f^{\prime \prime}(0) = 0 \)
- [tex]\( f^{\prime \prime}(1) = 4.4146 \)[/tex]
