(2) If $$ \mathrm{f}(x)=\left\{\begin{array}{l}x^{2}-2 x, x \geq 1 \ 3 x-4, x

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(2) If $$ \mathrm{f}(x)=\left\{\begin{array}{l}x^{2}-2 x, x \geq 1 \ 3 x-4, x<1\end{array}ight. $$ Discus the differentiability of $$ \mathrm{f}(x) $$ at $$ \mathrm{x}=1 $$

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1. **Determine the continuity at $$x=1$$:**  
   - For $$x \geq 1$$,  
     $$
     f(x) = x^2 - 2x.
     $$  
     Evaluating at $$x = 1$$:  
     $$
     f(1) = 1^2 - 2(1) = 1 - 2 = -1.
     $$
     
   - For $$x < 1$$,  
     $$
     f(x) = 3x - 4.
     $$  
     The left-hand limit as $$x \to 1^{-}$$:  
     $$
     \lim_{x \to 1^-} f(x) = 3(1) - 4 = 3 - 4 = -1.
     $$
     
   Since the left-hand limit and the value at $$x=1$$ are the same, $$f(x)$$ is continuous at $$x=1$$.

2. **Check differentiability at $$x=1$$:**  
   A function is differentiable at a point if the left-hand derivative and the right-hand derivative at that point are equal.
   
   - **Right-hand derivative ($$x \geq 1$$):**  
     For $$x \geq 1$$,  
     $$
     f(x) = x^2 - 2x.
     $$  
     Differentiate with respect to $$x$$:  
     $$
     f'(x) = 2x - 2.
     $$  
     At $$x = 1$$:  
     $$
     f'(1^+) = 2(1) - 2 = 0.
     $$
   
   - **Left-hand derivative ($$x < 1$$):**  
     For $$x < 1$$,  
     $$
     f(x) = 3x - 4.
     $$  
     Differentiate with respect to $$x$$:  
     $$
     f'(x) = 3.
     $$  
     At $$x = 1$$:  
     $$
     f'(1^-) = 3.
     $$
   
   Since the right-hand derivative $$f'(1^+)=0$$ and left-hand derivative $$f'(1^-)=3$$ are not equal, $$f(x)$$ is not differentiable at $$x=1$$.

3. **Conclusion:**  
   Although $$f(x)$$ is continuous at $$x=1$$, it is not differentiable at $$x=1$$ because the derivatives from the left and the right do not match.
At $$x=1$$, the function $$f(x)$$ is continuous but not differentiable because the left-hand and right-hand derivatives are not equal.

(2) If \( \mathrm{f}(x)=\left\{\begin{array}{l}x^{2}-2 x, x \geq 1 \\ 3 x-4, x<1\end{array}\right. \) Discus the differentiability of \( \mathrm{f}(x) \) at \( \mathrm{x}=1 \)

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