Your answer is incorrect, and your reasoning is not mathematically sound. The Correct answer is **C **because $|x|^n$ is differentiable for every integer $n>1$.

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Answer would be A

Mod in the angle of cos would not impact its value but mod in the sin will change the value of sin and x^3 and mod X is not differentiable at zero and as per the same reason if we continue the question then we will find that sin mod X and mod x^3 is not differentiable at zero proving the for the given function to remain differentiable A1 and A2 must be zero

Mod in the angle of cos would not impact its value but mod in the sin will change the value of sin and x^3 and mod X is not differentiable at zero and as per the same reason if we continue the question then we will find that sin mod X and mod x^3 is not differentiable at zero proving the for the given function to remain differentiable A1 and A2 must be zero