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For the function [tex]f(x) = 3 e^{-x^4}[/tex], find [tex]f''(x)[/tex]. Then find [tex]f''(0)[/tex] and [tex]f''(1)[/tex].

[tex]f''(x) = 12 x^2\left(4 x^4 - 3\right) e^{-x^4}[/tex]

Select the correct choice below and fill in any answer boxes in your choice.

A. [tex]f''(0) = 0[/tex] (Simplify your answer.)
B. [tex]f''(0)[/tex] is undefined.

Select the correct choice below and fill in any answer boxes in your choice.

A. [tex]f''(1) = 4.4146[/tex]
(Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places as needed.)
B. [tex]f''(1)[/tex] is undefined.



Answer :

GinnyAnswer
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]

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