We define the modulus function piecewise as :
F(x) = -(2 - 3x) = 3x - 2 , for x < 2/3 and
= 2 - 3x for x >= 2/3
So in order to check for differentiability , we check for left derivative and right derivative..
Here left derivative is w.r.t F(x) = 3x - 2..
So d(F(x)) / dx = 0 ==> d/dx(3x - 2) = 3 ..Hence left derivative is 3 at x = 3 [In fact at any point for x <= 2/3]
Now coming to right derivative , we have : F(x) = 2 - 3x
So d(F(x)) / dx = 0 ==> d/dx(2 - 3x) = -3..Hence right derivative is -3 at x = 3 [In fact at any point x >= 2/3]
As at x = 2/3 , Left derivative != Right derivative , so we can conclude
F(x) is not differentiable at x = 2/3