Let `f(x)` be a function defined on `(-a ,a)` with `a > 0.` Assume that `f(x)` is continuous at `x=0a n d(lim)_(xvec0)(f(x)-f(k x))/x=alpha,w h e r ek in (0,1)` then `f^(prime)(0^+)=0` b. `f^(prime)(0^-)=alpha/(1-k)` c. `f(x)` is differentiable at `x=0` d. `f(x)` is non-differentiable at `x=0`

Options

A. `f'(0^(+))=0`
B. `f'(0^(-))=(alpha)/(1-k)`
C. f(x) is defferentiable at x = 0
D. f(x) is non-differentiable at x = 0

Answer

B, C, D

Explanation

`because" "underset(xrarr0)(lim)(f(x)-f(lalpha))/(x)=alpha`
`rArr" "underset(xrarr0)(lim)(f(x)-f(0)+f(0)-(kx))/(x)=alpha`
`rArr" "underset(xrarr0)(lim)((f(x)-f(0))/(x)-(f(lx)-f(0))/(x))=alpha`
`rArr" "(underset(xrarr0)(lim)(f(x)-f(0))/(x))-(underset(xrarr0)(lim)(f(kx)-f(0))/(kx))k=alpha`
`rArr" "{{:(underset(xrarr0^(-))(lim)(f(x)-f(0))/(x)-underset(xrarr0^(-))(lim)(f(kx)-f(0))/(kx)k=alpha),(underset(xrarr0^(+))(lim)(f(kx)-f(0))/(kx)-underset(xrarr0^(+))(lim)(f(kx)-f(0))/(kx).k=alpha):}`
`={{:(f'(0^(-))-kf'(0^(-))=alpha),(f'(0^(+))-kf'(0^(+))=alpha):}`
`={{:((1-k)f'(0^(-))=alpha),((1-k)f'(0^(+))=alpha):}`
`={{:(f'(0^(-))=(alpha)/(-k)),(f'(0^(+))=(alpha)/(1-k)):}`
`therefore" "f'(0)=f'(0^(-))=f'(0^(+))=(alpha)/(1-k)`