Let `f(x) = x^(3) - x^(2) + x + 1 and g(x) = {{:(max f(t)",", 0 le t le x,"for",0 le x le 1),(3-x",",1 lt x le 2,,):}` Then, g(x) in [0, 2] is

Options

A. continuous and differentiable on [0,2]
B. continuous but not differentiable on [0,2]
C. neither continuous nor differentiable on [0,2]
D. none of these

Answer

B

Explanation

We have
`f(x)=x^(3)-x^(2)+x+1`
`f'(x)=3x^(2)-2x+1gt 0"for all x"" "[therefore "Disc"lt 0 and "Ceff of "x^(2) gt 0]`
`Rightarrow` f(x) is increasing for all `x in R`
`therefore g(x)={{:(,x^(3)-x^(2)+x+1,0 le x le 1),(,3-x,1 lt x le 2):}`
Clearly, g(x) is everywhere continuous and differentiable except possible at x=1
`"Clearly" underset(x to 1^(-))lim g(x)=underset(x to 1^(+))lim g(x)=g(1)`
So, it is continuous at x=1
We observe that
`("LHD at x=1")={(d)/(dx)(x^(3)-x^(2)+x+1)}_(x=1)==(3x^(2)-2x+1)_("at x=1")=2`
Clearly, g(x) is not differentiable at x=1