Let `f (x) = x^3-x^2-x+1` and `g(x) = {max{f(t); 0<=t<=x}, 0<=x<=1, 3-x, 1<=x<=2` Discuss the continuity and differentiability of the function g (x) in the interval (0, 2).

Answer

g (x) is continuous for all ` x in (0, 2) -{1] and g(x) ` is differentiable for all ` x in (0, 2) - {1}`

Explanation

Given, ` f(x) = x^(3) - x^(2) - x + 1 `
` rArr" " f' (x) = 3x^(2) - 2x -1 = (3x + 1)(x-1)`
`:. ` f (x) is increasing for ` x in (-infty, -1//3) cup (1, infty)` and decreasing for ` x in (-1//3, 1)`
Also, given `g (x) ={{:(max {f(t),0le t le x}", " 0 le x le 1),(" 3-x, "1 lt x le 2):}`
`rArr g(x) = {{:(g(x)", " 0 le x le 1),(3-x", " 1 lt x le 2):}`
` rArr g(x) ={{:(x^(3)-x^(2)-x+1", " 0 le xle 1),(" 3-x, " 1 lt x le 2):}`
At x = 1,
RHL = `underset( xto 1) lim (3-x) = 2 `
and LHL =` underset( x to 1) lim (x^(3)-x^(2) - x +1) = 0`
`:. ` It is discontinuous at x = 1.
Also, ` g'(x) = {{:(3x^(2)-2x-1", " 0 le x le 1),(" -1, " 1 lt x le 2 ):}`
`rArr" " g'(1^(+)) =- 1`
and ` g'(1^(-)) = 3 - 2=0`
`:.` g(x) is continuous for all ` x in (0, 2) - {1} and g (x) ` is differentiable for all ` x in (0, 2) - {1} ` .