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The map $f(x) = a_0 \cos \mid x \mid +a_1 \sin \mid x \mid +a_2 \mid x \mid ^3$ is differentiable at $x=0$ if and only if

1. $a_1=0$ and $a_2=0$
2. $a_0=0$ and $a_1=0$
3. $a_1=0$
4. $a_0, a_1, a_2$ can take any real value

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