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Evaluate [tex]\( f(3) \)[/tex] given that [tex]\( F(x) = 12 \ln(x) + 1 \)[/tex] is an antiderivative of [tex]\( f(x) \)[/tex].

Sagot :

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]