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

$\Rightarrow g \left(x\right)=\frac{f \left(x\right)}{x}$
$\Rightarrow g'\left(x\right)=\frac{xf'\left(x\right)-f \left(x\right)}{x^{2}}$
$g'\left(c^{+}\right)=\displaystyle\lim_{h\to0} \frac{\left(c+h\right)f'\left(c+h\right)-f\left(c\right)}{h}$
$\left[f' \left(c+h\right)<\,0\right]$
$g'\left(c^{+}\right)<\,0$