Answer is: option3
\( -\frac{1}{2}x^2 + 2x - 1 \)Solution:
We are given:
- The slope of the curve at each point \( (x, y) \) is \( -x + 2 \), meaning the derivative of the function \( f(x) \) is:
\[ f'(x) = -x + 2 \]
- The curve passes through the point \( (2, 1) \), which gives us the initial condition \( f(2) = 1 \).
\[ f(x) = \int (-x + 2)\, dx = -\frac{1}{2}x^2 + 2x + C \]
Use the point \( (2, 1) \) to solve for \( C \)
\[ 1 = -\frac{1}{2}(2)^2 + 2(2) + C \]
\[ 1 = -2 + 4 + C \]
\[ 1 = 2 + C \Rightarrow C = -1 \]
So, the equation of the curve is:
\[ f(x) = -\frac{1}{2}x^2 + 2x - 1 \]
This matches option (C).