Given: The correct function is f(x)=(x²-1) mod( x² - 3x + 2 ) +cos mod( x )

To find: The given function is not differential at?

Solution:

• Now lets WRITE the given function:

              f(x)=(x²-1) mod( x² - 3x + 2 ) + cos mod( x )

• So now, let us consider two FUNCTIONS: G(x) and H(x)

             g(x) = (x²-1) mod( x² - 3x + 2 )

             h(x) = cos mod( x )

• So eventually h(x) becomes:

            h(x) =  g(x) + h(x)

• Now consider the function h(x) = cos mod( x )

• As we know that mod(x) is x when x≥0 and -x when x<0

• So cos mod(x) will be cos x when x≥0 and cos(-x) when x<0

• But we know that cos(-x) = cos x,

                        so it becomes cos x ∈ R. As the graph of cos x do not have any SHARP point so it is always differentiable.

• Now lets consider g(x) = (x²-1) mod( x² - 3x + 2 )

• Solving this,

                g(x) = (x-1)(x+1) mod((x-1)(x-2))

                g(x) = (x-1)(x+1) mod(x-1)mod(x-2)

• So the critical points are: 1, -1, 2

             When x≤-1, g(x) will be positive.

             When -1

             When 1

             When x>2, g(x) will be positive.

• So we can see that graph is not changing in third and fourth case, so the function is not differentiable at 2.

Answer:

             So the function is not differentiable at 2.