To find [tex]\( f(x) \)[/tex] and evaluate [tex]\( f(3) \)[/tex], follow these steps:
1. Identify [tex]\( F(x) \)[/tex]:
The function given is [tex]\( F(x) = 12 \ln(x) + 1 \)[/tex]. This function is the antiderivative of [tex]\( f(x) \)[/tex].
2. Differentiate [tex]\( F(x) \)[/tex]:
To find [tex]\( f(x) \)[/tex], we need to take the derivative of [tex]\( F(x) \)[/tex].
- Recall the property of derivatives: [tex]\(\frac{d}{dx} [\ln(x)] = \frac{1}{x}\)[/tex].
- Using the constant multiple rule and the sum rule, differentiate [tex]\( F(x) \)[/tex] term by term:
[tex]\[
F(x) = 12 \ln(x) + 1
\][/tex]
[tex]\[
\frac{d}{dx}[F(x)] = \frac{d}{dx}[12 \ln(x)] + \frac{d}{dx}[1]
\][/tex]
- The derivative of [tex]\( 12 \ln(x) \)[/tex] is [tex]\( 12 \cdot \frac{1}{x} = \frac{12}{x} \)[/tex].
- The derivative of the constant [tex]\( 1 \)[/tex] is [tex]\( 0 \)[/tex].
Therefore,
[tex]\[
f(x) = \frac{12}{x}
\][/tex]
3. Evaluate [tex]\( f(3) \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] into the expression for [tex]\( f(x) \)[/tex]:
[tex]\[
f(3) = \frac{12}{3}
\][/tex]
Simplifying this gives:
[tex]\[
f(3) = 4
\][/tex]
The final answers are:
[tex]\[
f(x) = \frac{12}{x}
\][/tex]
[tex]\[
f(3) = 4
\][/tex]
