Check the continuity of f(x) = `{{:(x^(2)/2, if 0le x le 1),(2x^(2)-3x+3/2, if 1 lt x le 2):}` at `x = 1`

Options

A. f, f' and f'' are continuous in [0,2]
B. f and f' are continuous in [0,2] whereas f'' is continuous in `[0,1] uu (1,2]`
C. f,f' and f'' are continuous in `[0,1) uu (1,2]`
D. none of these

Answer

A

Explanation

Clearly, f is continuous and differentiable at x=1, Thus f(x) is continuous and differentiable on [0,2]
`therefore f'(x)={{:(,x,0 le x lt 1),(,4x-3,1 le x le 2):}`
We observe that f'(x) is continuous on [0,2] but it is not differentiable at x=1,
`therefore f''(x)={{:(,1,0 le x lt 1),(,4,1 lt x le 2):}`
Clearly, f''(x) is not continuous at x=1
Hence, f and f'' are continuous on [0,2] whereas f'' is continuous on `[0,1) uu (1,2]`