Let's find the derivatives and evaluations step by step for the function [tex]\( f(x) = 2x^4 - 3e^x \)[/tex].
1. First Derivative [tex]\( f'(x) \)[/tex]
To find the first derivative of [tex]\( f(x) \)[/tex], we need to differentiate each term individually:
[tex]\[
f(x) = 2x^4 - 3e^x
\][/tex]
The derivative of [tex]\( 2x^4 \)[/tex] is [tex]\( 8x^3 \)[/tex] and the derivative of [tex]\( -3e^x \)[/tex] is [tex]\( -3e^x \)[/tex]. Thus, the first derivative is:
[tex]\[
f'(x) = 8x^3 - 3e^x
\][/tex]
2. Evaluation of the First Derivative at [tex]\( x = 1 \)[/tex]
Next, we substitute [tex]\( x = 1 \)[/tex] into the first derivative to find [tex]\( f'(1) \)[/tex]:
[tex]\[
f'(1) = 8(1)^3 - 3e^{1}
\][/tex]
Since [tex]\( 1^3 \)[/tex] is [tex]\( 1 \)[/tex] and [tex]\( e^1 \)[/tex] is [tex]\( e \)[/tex], we get:
[tex]\[
f'(1) = 8 - 3e
\][/tex]
3. Second Derivative [tex]\( f''(x) \)[/tex]
To find the second derivative, we differentiate the first derivative [tex]\( f'(x) = 8x^3 - 3e^x \)[/tex]:
The derivative of [tex]\( 8x^3 \)[/tex] is [tex]\( 24x^2 \)[/tex] and the derivative of [tex]\( -3e^x \)[/tex] remains [tex]\( -3e^x \)[/tex]. Thus, the second derivative is:
[tex]\[
f''(x) = 24x^2 - 3e^x
\][/tex]
4. Evaluation of the Second Derivative at [tex]\( x = 1 \)[/tex]
Finally, we substitute [tex]\( x = 1 \)[/tex] into the second derivative to find [tex]\( f''(1) \)[/tex]:
[tex]\[
f''(1) = 24(1)^2 - 3e^{1}
\][/tex]
Since [tex]\( 1^2 \)[/tex] is [tex]\( 1 \)[/tex] and [tex]\( e^1 \)[/tex] is [tex]\( e \)[/tex], we get:
[tex]\[
f''(1) = 24 - 3e
\][/tex]
Therefore, the answers are:
[tex]\[
\begin{array}{l}
f^{\prime}(x) = 8x^3 - 3e^x \\
f^{\prime}(1) = 8 - 3e \\
f^{\prime \prime}(x) = 24x^2 - 3e^x \\
f^{\prime \prime}(1) = 24 - 3e
\end{array}
\][/tex]
