Question

Let f(x) `={underset( 3 sin x " " x le 0)(X^(3) +X^(2) -10X " " X lt0).` Examine the behaviour of f(x) at =0

Answer 1

f(x) is continuous at x=0
`f(x) ={underset( 3 cos x " "x gt 0)(3x^(2) +2x -10 x lt 0).`
f(0) =3 and f(0) =- 10 thus f(x) is non- differentiable at `x=0" "rArr " "x =0 ` is a critical point.
Also derivative change sigh from negative to positive , so x=0 is a point of local minima.