The following figure shows the graphs of a differentiable fucntion `y = f(x)` on the interval `[a,b]` (not contaning 0).

Let ` g(x) = f(x)//x` which of the following is a possible graph of `y =g(x)` ?

Options

A.
B.
C.
D.

Answer

B

Explanation

`f'(c ) = 0 f'(c^(-)) gt 0 f'(c^(+)) lt 0`
` g(x) = (f(x))/(x) g'(x) = (xf'(x)-1)/(x^(2))`
`g'(c^(+)) = underset(hrarr0)("lim")((c+h)f'(c+h)-1)/((c+h)^(2)) lt 0 , (f'(c+h) lt 0)`
hence fig. (2)