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Answer is Right!
Answer is Wrong!
The correct answer is $\boxed{\text{A}. x^{-1}}$.
The slope of the tangent line to the curve at any point $(x, y)$ is $y'(x)$. The $y$-axis intercept of the tangent line is $y'(x)x$. We are given that $y'(x)x = -y$. This means that $y'(x) = -\frac{y}{x}$.
The derivative of $x^{-1}$ is $-\frac{1}{x^2}$. Therefore, $y'(x) = -\frac{y}{x}$ is equivalent to $y = -\frac{1}{x^2}x$. This can be rewritten as $y = -\frac{1}{x}$.
Therefore, $f(x)$ is proportional to $x^{-1}$.
Here is a brief explanation of each option:
- Option A: $x^{-1}$. This is the correct answer. The derivative of $x^{-1}$ is $-\frac{1}{x^2}$. Therefore, $y'(x) = -\frac{y}{x}$ is equivalent to $y = -\frac{1}{x}$. This can be rewritten as $y = -\frac{1}{x^2}x$.
- Option B: $x^2$. This is not the correct answer. The derivative of $x^2$ is $2x$. Therefore, $y'(x) = -\frac{y}{x}$ is not equivalent to $y = -2x$.
- Option C: $x^3$. This is not the correct answer. The derivative of $x^3$ is $3x^2$. Therefore, $y'(x) = -\frac{y}{x}$ is not equivalent to $y = -3x^2$.
- Option D: $x^4$. This is not the correct answer. The derivative of $x^4$ is $4x^3$. Therefore, $y'(x) = -\frac{y}{x}$ is not equivalent to $y = -4x^3$.