Suppose that \( f(x)=\displaystyle\frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}},\ x\gt 1 \), then which of the following statements are TRUE?
(A) \( \displaystyle\frac{df(x)}{dx}=2x \)
(B) \( f \) is a decreasing function on \( (1,+\infty) \)
(C) \( f \) is an increasing function on \( (1,+\infty) \)
(D) \( f \) has a local minima at \( x=2 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (C) only
Since \( e^{k\log x}=e^{\log x^k}=x^k \), the function simplifies to a polynomial:
\( \displaystyle f(x)=\frac{x^5-x^4}{x^3-x^2} \)
\( \displaystyle =\frac{x^4(x-1)}{x^2(x-1)} \)
\( \displaystyle =x^2 \)
Cancelling \( (x-1) \) is allowed because x > 1. So
\( \displaystyle f'(x)=2x \)
\( \displaystyle 2x\gt 0\ \text{for}\ x\gt 1 \)
So f(x) = x2, f′(x) = 2x, and f′(x) > 0 on the whole interval \( (1,+\infty) \).
Checking the statements
(A) \( \frac{df(x)}{dx}=2x \) — Correct. f(x) = x2, so f′(x) = 2x.
(B) f is a decreasing function on \( (1,+\infty) \) — Incorrect. f′(x) = 2x > 0 there.
(C) f is an increasing function on \( (1,+\infty) \) — Correct. f′(x) = 2x > 0 for every x > 1.
(D) f has a local minima at x = 2 — Incorrect. f′(2) = 4 ≠ 0, and f keeps increasing through x = 2.