Question

Which of the following statements is always true?

• A. If $f(x)$ is decreasing, then $\frac{1}{f(x)}$ is increasing
• B. If $f(x)$ is decreasing, then $\frac{1}{f(x)}$ is also decreasing
• C. If both $f$ and $g$ are positive functions such that $f$ is decreasing and $g$ is increasing, then $\frac{f}{g}$ is a decreasing function
• D. If both $f$ and $g$ are positive functions such that $f$ is increasing and $g$ is decreasing, then $\frac{f}{g}$ is a decreasing function

Hint:

Think of it logically: if a fraction's numerator is shrinking (decreasing) while its denominator is simultaneously expanding (increasing), the overall value of the fraction must drop rapidly. Both actions work together to make the function decrease!

Answer 1

The Correct Option is C

Concept: The derivative of a quotient function $\frac{f(x)}{g(x)}$ tells us its interval monotonicity behavior. Using the standard Calculus quotient rule: $$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$$ By verifying the absolute sign of the numerator components under given conditions, we can systematically determine if a function is strictly increasing or decreasing across its domain. Step 1: Analyze statements (A) and (B).
Let us find the derivative of $y = \frac{1}{f(x)}$ using the chain rule: $$y' = -\frac{f'(x)}{[f(x)]^2}$$ If $f(x)$ is a decreasing function, we know that $f'(x) < 0$. This forces the numerator term $-f'(x)$ to be strictly positive ($>0$). However, if $f(x)$ changes sign across its domain (for example, passing from positive values to negative values through a zero point), the function breaks down into separate discontinuous asymptotic branches. Therefore, statements (A) and (B) are not always universally true.

Step 2: Analyze statement (C) using the Quotient Rule.
Let our composite target fraction function be defined as $h(x) = \frac{f(x)}{g(x)}$. Differentiating both sides with respect to $x$: $$h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$$ Now, let us audit the individual mathematical signs of each parameter based on the precise parameters given in choice (C):
• $f(x)$ and $g(x)$ are both positive functions $\implies f(x) > 0, \ g(x) > 0$
• $f(x)$ is a strictly decreasing function $\implies f'(x) < 0$
• $g(x)$ is a strictly increasing function $\implies g'(x) > 0$

Step 3: Evaluate the net sign of the derivative numerator.
Substitute the individual sign profiles into the numerator subtraction terms:
• First term: $f'(x)g(x) \rightarrow (-) \times (+) = (-)$
• Second term: $-f(x)g'(x) \rightarrow - \big((+) \times (+)\big) = (-)$ Since the numerator is composed of two combined negative numbers, the net value is guaranteed to be strictly less than zero: $$\text{Numerator} < 0 \quad \text{and} \quad \text{Denominator } [g(x)]^2 > 0 \quad \Rightarrow \quad h'(x) < 0$$ Because the first derivative is always negative, the rational function $h(x) = \frac{f}{g}$ is strictly decreasing, confirming that statement (C) is a mathematically robust universal rule.