If 1`f(x)={x^(2),xle0`.Investigate the functions at x for maxima/manima

Explanation

Clearly `f(x) =f(x^(+))=underset(xrarr0)lim (2sinx)=0^(+)`
Also `f(0^(-))=underset(xrarr0)lim x^(2)=0^(+)`
Thus f(x) is continous at x=0
`f(0)ltf(0+delta)and f(0)ltf(0-delta)`
Thus x=0 is point of minima
We can draw the graph of the function to verify this

Also f(x) =`{2x,xle0`
`2cos x xge0`
`f(0-delta)lt0` and `f(0+delta)gt0`
Thus derivative changes sign from negative to positive with f(0)=0
So x=0 is point of minima.