The critical points of `f(x)=(|2-x|)/(x^2) is//are`

Options

A. x=0,2
B. x=2,4
C. x=2,-4
D. none of these

Answer

D

Explanation

We have
`f(x)=(|2-x|)/(x^2)={{:((x-2)/x^2", " xge2),((2-x)/x^2"," if xlt2):}`
`rArrf'(x)={{:(-1/(x^2)+4/x^3"," ifxgt2),(1/x^2-4/x^3", "if xlt2):}`
`rArrf'(x)={{:((4-x)/x^3", "xgt2),((x-4)/x^3", "xlt2):}`
Clearly , f'(x)=0 at x=4 and f'(x) changes its sign from positive to negative in the nighbourhood af x=4 Therefore x=4 is a point of maximum
Also, `f(x)rarr0 " as " xrarr2 " and " f(x)gt0 " for all " x ne 2`
Therefore , x=2 is a point of local minimum
Clearly , f(x)is not differentiable at x=0.
Hence x=0,2,4 are critical points of f(x)