Let: `f(x)=int_0^x|2t-3|dtdot` Then discuss continuity and differentiability of `f(x)at x=3/2`

Explanation

`f(x)=int_(0)^(x)|2t-3|dt`
`={(int_(0)^(x)(3-2t)dt, xlt3/2),(int_(0)^(3//2)(3-2t)dt+int_(3//2)^(x)(2t-3)dt, xge3/2):}`
`={(3x-x^(2),xlt3/2),(x^(2)-3x+9/2,xge3/2):}`
Clearly `f(x)` is continuous at `x=3/2` as `f((3^(-))/2)=f'((3^(+))/2)=9/4`
`f'(x)={(3-2x,xlt3/2),(2x-3,xgt3/2):}`
`:.f'((3^(-))/2)=f'((3^(+))/2)=0`
Hence `f(x)` is differentiable at `x=3/2`.