The function `{{:(|x-3|,,ifxge1),((x^(2))/(2)-(3x)/(2)+(13)/(4),, if xlt 1):}` is

Options

A. continuous and differentiable at x = 3
B. continuous at x = 3, but not differentiable at x = 3
C. continuous and differentiable everywhere
D. continous at x = 1, but not differentiable at x = 1

Answer

B

Explanation

Clearly, f(x) is not differentialbe at x = 3.
Now, `lim_(x to 3-) f(x) = lim_(h to 0) f(3-h)`
` = lim_(h to 0)|3-h-3| = 0`
`lim_(x to 3+) f(x) = lim_(h to0) f(3+h)`
` = lim_(h to 0) | 3 + h -3| = 0`
and `f(3) = |3-3| = 0`
`therefore f(x)` is continuous at x = 3.