Let f(x) be a function for all x `in` R and f'(0) = 1 then g(x)= f (lxl)-`sqrt((1-cos2x)/2)`
| A. |
is differentiable at x : 0 and its value is 1 |
| B. |
is differentiable at x : 0 and its value is 0 |
 |
| C. |
is non differentiable at x: 0 as its graph has sharp turn at x: 0 |
| D. |
is non differentiable at x: 0 as its graph has vertical tangent at x: 0 |
g(x)=f(|x|)-|sin x|
`g^1(0^+)=lim_(h->0) (f(|h|)-|sinh|-f(0))/h` `=lim_(h->0) (f(h)-f(0))/h-lim_(h->) (sinh)/h=1-1=0`
`g^1(0^-)=lim_(h->0) (f(|-h|)-|sin(-h)|-f(0))/-h =lim_(h->0) (f(h)-f(0))/-h + lim_(h->0) (sinh)/h=-1+1=0`
Thus g(x) is differentiable at x=0 and `g^1`(0)=0