Question

assignment 10: problem 1
(1 point)
find f if f(x) = 2/x, x < 0 and f(-1) = 4.
f(x) = \square

Explanation:

Step1: Integrate \( f'(x) \)

To find \( f(x) \), we integrate \( f'(x) = \frac{2}{x} \) with respect to \( x \). The integral of \( \frac{1}{x} \) is \( \ln|x| \), so:

$$ f(x) = \int \frac{2}{x} dx = 2\ln|x| + C $$

Step2: Determine the constant \( C \)

We know \( f(-1) = 4 \) and \( x < 0 \), so \( |x| = -x \). Substitute \( x = -1 \) into \( f(x) \):

$$ 4 = 2\ln|-1| + C $$

Since \( \ln|-1| = \ln(1) = 0 \), we have:

$$ 4 = 2(0) + C \implies C = 4 $$

Step3: Write the final function

Substitute \( C = 4 \) back into \( f(x) \). For \( x < 0 \), \( |x| = -x \), but \( \ln|x| = \ln(-x) \) when \( x < 0 \). So:

$$ f(x) = 2\ln(-x) + 4 $$

(Note: Since \( x < 0 \), \( -x \) is positive, so the logarithm is defined.)

Answer:

\( 2\ln(-x) + 4 \)