State whether the following statements are True of False.
If `f(x)=x^(2)-3x+1`, then derivative of f(x) at x=2, f'(2)=1.

Answer

To determine whether the statement "If \( f(x) = x^2 - 3x + 1 \), then the derivative of \( f(x) \) at \( x = 2 \), \( f'(2) = 1 \)" is true or false, we will follow these steps:

### Step 1: Find the derivative of \( f(x) \)

The function given is:
\[ f(x) = x^2 - 3x + 1 \]

To find the derivative \( f'(x) \), we differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(x^2) - \frac{d}{dx}(3x) + \frac{d}{dx}(1) \]

Calculating each term:
- The derivative of \( x^2 \) is \( 2x \).
- The derivative of \( 3x \) is \( 3 \).
- The derivative of a constant \( 1 \) is \( 0 \).

Thus, combining these results, we have:
\[ f'(x) = 2x - 3 \]

### Step 2: Evaluate the derivative at \( x = 2 \)

Now we need to find \( f'(2) \):
\[ f'(2) = 2(2) - 3 \]

Calculating this gives:
\[ f'(2) = 4 - 3 = 1 \]

### Step 3: Compare with the given statement

The statement claims that \( f'(2) = 1 \). Since we calculated \( f'(2) \) and found it to be \( 1 \), the statement is indeed true.

### Conclusion

Thus, the statement is **True**.

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