Dangerous Derivative dudes

Calculus Level pending

If f ( x ) = sin ⁡ ( 2 π ⌊ π 2 x ⌋ ) 5 + [ x 2 ] f(x) = \dfrac{\sin(2\pi\lfloor \pi^2x\rfloor )}{5+[x^2]} , then which of the following is true?

Notation : ⌊ ⋅ ⌋ \lfloor \cdot \rfloor denotes the floor function .

f ′ ( x ) f'(x) exist for all x x , but f ′ ′ ( x ) f''(x) doesn't exist for some x x f ( x ) f(x) is discontinuous at some x x f ′ ′ ( x ) f''(x) exist for all x x f ( x ) f(x) is continuous for all x x , but the derivative f ′ ( x ) f'(x) doesn't exist for same x x

Danish Ahmed by Danish Ahmed , 24, India

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