Applications of Derivatives
Monotonicity via Second Derivative
nta_pyq_2024_jan
Grade 12
Question:
Let $g(x)=3f\left(\dfrac{x}{3}\right)+f(3-x)$ and $f''(x)>0$ for all $x\in(0,3)$. If $g$ is decreasing in $(0,\alpha)$ and increasing in $(\alpha,3)$, then $8\alpha$ is:
Step-by-Step Solution
Key Concept: $g'(x)=f'(x/3)-f'(3-x)$. Since $f''>0$, $f'$ is strictly increasing. $g'(x)<0\Rightarrow f'(x/3)<f'(3-x)\Rightarrow x/3<3-x\Rightarrow x<9/4$. So $\alpha=9/4$ and $8\alpha=18$.
$g'(x)=f'(x/3)-f'(3-x)$. $f''>0\Rightarrow f'$ increasing. $g'(x)<0\Rightarrow x/3<3-x\Rightarrow 4x/3<3\Rightarrow x<9/4$. So $\alpha=9/4$, $8\alpha=18$.
Correct Answer: 3