Question

The following figure shows the graph of a differentiable function $y=f(x)$ on the interval $[a,b]$ (not containing $0)$

Let $g\left(x\right)=\frac{f\left(x\right)}{x}$ Which of the following is a possible graph of $y=g(x)$ ?

• A. Fig 1
• B. Fig 2
• C. Fig 3
• D. Fig 4

Answer 1

Answer: (B)

Let $f^{\prime }(x)=0$ at $c$
$\therefore f^{\prime }(c)=0$
$f^{\prime }(c^{-})>0$
$\Rightarrow f^{\prime }(c^{+})<0$

$\Rightarrow g\left(x\right)=\frac{f\left(x\right)}{x}$
$\Rightarrow g^{\prime }\left(x\right)=\frac{x{f}^{\prime }\left(x\right)-f\left(x\right)}{x^2}$
$g^{\prime }\left(c^{+}\right)=\underset{h\to 0}{\lim }\frac{\left(c+h\right)f^{\prime }\left(c+h\right)-f\left(c\right)}{h}$
$\left[f^{\prime }\left(c+h\right)<0\right]$
$g^{\prime }\left(c^{+}\right)<0$