Let [g(x)=3 f((x)/(3))+f(3-x)] and [f^ (x)0] for all [x (0,3)].

Question

Let $$g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$$ and $$f^{\prime \prime}(x)>0$$ for all $$x \in(0,3)$$. If $g$ is decreasing in $$(0, \alpha)$$ and increasing in $$(\alpha, 3)$$, then $$8 \alpha$$ is :

Accepted Answer

$$g(x)=3 f\left(\frac{x}{3}\right)+f(3-x) \text { and } f^{\prime \prime}(x)>0 \forall x \in(0,3)$$

$$\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})$$ is increasing function

$$\begin{aligned}
& g^{\prime}(x)=3 \times \frac{1}{3} \cdot f^{\prime}\left(\frac{x}{3}\right)-f^{\prime}(3-x) \
& =f^{\prime}\left(\frac{x}{3}\right)-f^{\prime}(3-x)
\end{aligned}$$

If $$\mathrm{g}$$ is decreasing in $$(0, \alpha)$$

Solutions · 1