Answer is: option1
\( -3 \)Solution:
The equation of the tangent line at \( x = 2 \) is given by:
\[ y = f(2) + f'(2)(x - 2) \]
Substituting the given values \( f(2) = \frac{5}{2} \) and \( f'(2) = \frac{1}{2} \):
\[ y = \frac{5}{2} + \frac{1}{2} (x - 2) \]
To approximate a root of \( f(x) \), we set \( y = 0 \):
\[ 0 = \frac{5}{2} + \frac{1}{2} (x - 2) \]
1. Subtract \( \frac{5}{2} \) from both sides:
\[ -\frac{5}{2} = \frac{1}{2} (x - 2) \]
2. Multiply both sides by 2 to clear the fraction:
\[ -5 = x - 2 \]
3. Solve for \( x \):
\[ x = -3 \]
The answer is \( -3 \).