Applications of Derivatives
Monotonicity and Injectivity
nta_pyq_2023_jan
Grade None
Question:
Let $f:(0,1)\to\mathbb{R}$ be a function defined by $f(x)=\dfrac{1}{1-e^{-x}}$, and $g(x)=(f(-x)-f(x))$. Consider two statements: (I) g is an increasing function in $(0,1)$; (II) g is one-one in $(0,1)$. Then,
Only (I) is true
Only (II) is true
Neither (I) nor (II) is true
Both (I) and (II) are true
Step-by-Step Solution
Key Concept: $g(x)=f(-x)-f(x)=\frac{1}{1-e^x}-\frac{1}{1-e^{-x}}=\frac{1+e^x}{1-e^x}$. Compute $g'(x)=\frac{2e^x}{(1-e^x)^2}>0$ for all $x\in(0,1)$.
$g(x)=\frac{1+e^x}{1-e^x}$, $g'(x)>0$. Both statements are true.
Correct Answer: 4