Let `f(x)={{:(0,x lt 0),(x^(2),xge0):}`, then for all values of x

Options

A. f is continous and differentiable
B. f is differentiable but not continous
C. f is continous but not differentiable
D. none of the above

Answer

A

Explanation

Given `f(x)={{:(0,x lt 0),(x^(2),x ge0):}`
`therefore underset(x to 0)(lim)f(x)=underset(x to 0)(lim)x^(2)=0`
annd f(0)=0
`thereforef(x)` is continuous at x=0
Now, `Lf'(x)=underset(xto0^(-))(lim)(f(x)-f(0))/(x-0)`
`=underset(hto0)(lim)(f(0-h)-0)/(-h)=0`
and `Rf'(x)=underset(xto0^(+))(lim)(f(x)-f(0))/(x-0)`
`=underset(hto0)(lim)((0+h)^(2)-0)/(h)=0`
`impliesLf'(x)=Rf'(x)`
`thereforef(x)` is differentiable at x=0