Let $$f(x) = a_0 + a_1 \mid x\mid +  a_2 \mid x\mid^2 + a_3 \mid x\mid^3$$, where $$a_0, a_1, a_2$$ and $$a_3$$ are constants. Which of the following statements is correct?

For the function to be differentiable at x=0, f'(x) =$$\lim\ h\longrightarrow\ 0$$  $$\ \frac{\ f\left(x+h\right)-f\left(x\right)}{h}$$

=> f'(0)=$$\lim\ h\longrightarrow\ 0$$ $$\ \frac{\ f\left(h\right)-f\left(0\right)}{h}$$

Now using the value of f(x), we get f'(0) =$$\lim\ h\longrightarrow\ 0$$   $$\ \frac{\ a_0+a_1\mid h\mid+a_2\mid h\mid^2+a_3\mid h\mid^3-a_0}{h}$$

Now, f'(0)= $$\lim\ h\longrightarrow\ 0$$  $$\ \frac{a_1\mid h\mid+a_2\mid h\mid^2+a_3\mid h\mid^3}{h}$$

For h<0, f'(0)=$$-\ a_1+a_2h-a_3h^2$$

For h>0, f'(0)=$$\ a_1+a_2h+a_3h^2$$

So for $$h\longrightarrow\ 0$$, the two values will only be equal if $$a_1$$ = 0.

Hence, C is the answer.