If `f'(x) = g(x) (x-a)^2`, where g(a) `!= 0` and g is continuous at x = a, then :

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