Applications of Derivatives
Local Maximum — Decreasing Interval of |t+1|/t²
nta_pyq_2026_jan
Grade 12
Question:
Let $(2\alpha,\alpha)$ be the largest interval in which the function $f(t)=\dfrac{|t+1|}{t^2}$, $t<0$, is strictly decreasing. Then the local maximum value of the function $g(x)=2\log_e(x-2)+\alpha x^2+4x-\alpha$, $x>2$, is _____
Step-by-Step Solution
Key Concept: For $t<0$: $f$ is decreasing on $(-2,-1)$ (check sign of $f'$ in each sub-interval). Largest decreasing interval $=(-2,-1)=(2\alpha,\alpha)\Rightarrow\alpha=-1$.
$\alpha=-1$. Local max of $g$ at $x=3$: $g(3)=4$.
Correct Answer: 4