Applications of Derivatives
Monotonicity via Differential Inequalities
nta_pyq_2023_apr
Grade 12
Question:
Let $f:[2,4]\to\mathbb{R}$ be a differentiable function such that $(x\log_e x)f'(x)+(\log_e x)f(x)+f(x)\geq 1,\ x\in[2,4]$ with $f(2)=\dfrac{1}{2}$ and $f(4)=\dfrac{1}{2}$. Consider the following two statements: (A) $f(x)\leq 1$, for all $x\in[2,4]$; (B) $f(x)\geq\dfrac{1}{8}$, for all $x\in[2,4]$. Then,
Neither statement (A) nor statement (B) is true
Only statement (B) is true
Both the statements (A) and (B) are true
Only statement (A) is true
Step-by-Step Solution
Key Concept: Rewrite the inequality as $\frac{d}{dx}[x\ln x\cdot f(x)]\geq 1=\frac{d}{dx}(x)$, so $h(x)=x\ln x\cdot f(x)-x$ is increasing on $[2,4]$.
$h(x)=x\ln x\cdot f(x)-x$ is increasing. $h(x)\leq h(4)=4\ln 4\cdot\frac{1}{2}-4=\ln 4-4\Rightarrow f(x)\leq 1$. $h(x)\geq h(2)=2\ln 2\cdot\frac{1}{2}-2=\ln 2-2\Rightarrow f(x)\geq\frac{1}{8}$. Both (A) and (B) are true.
Correct Answer: 3