If `f^(prime)x=g(x)(x-a)^2,w h e r eg(a)!=0,a n dg` is continuous at `x=a ,` then (a) `f` is increasing in the neighbourhood of `a` if `g(a)>0` (b)`f` is increasing in the neighbourhood of `a` if `g(a)<0` (c)`f` is decreasing in the neighbourhood of `a` if `g(a)>0` (d)`f` is decreasing in the neighbourhood of `a` if `g(a)<0`

Options

A. f is increasing in the neighborhood of a if `g(x)gt0`
B. f is increasing in the neighborhood of a if `g(x)lt0`
C. f is decreasing in the neighborhood of a if `g(x)gt0`
D. f is decreasing in the neighborhood of a if `g(x)lt0`

Answer

1,4

Explanation

Since `g(a) ne 0` either `g(a) gt0 or g(a) lt0`
Let `g(a)gt0` since g(X)is continous at x=a there exist a neighborhood of a in which `g(x)gt0`
Thus `f(X)gt0` Therefore f(X) is increasing in the neighborhood of a
Let `g(a) lt0` since g(x) is ocntinous at x=a there exist a neighborhood of a in which `g(x)lt0`
Thus `f(X)lt0` Therefore f(X) is decreasing in the neightborhood of a