Calculate Descending Area of f(x)=2x-x²+1: Quadratic Function Analysis

To determine where the function $f(x) = 2x - x^2 + 1$ is decreasing, we follow these steps:

• Step 1: Find the derivative
  The derivative of the function $f(x) = 2x - x^2 + 1$ is found using standard calculus rules. Thus, $f'(x) = \frac{d}{dx}(2x - x^2 + 1) = 2 - 2x$.
• Step 2: Determine where the derivative is negative
  Set the derivative $2 - 2x$ less than zero to find where the function is decreasing:

$2 - 2x < 0$

Solve for $x$:
Subtract 2 from both sides to get:

$-2x < -2$

Now, divide both sides by -2, remembering to reverse the inequality sign:

$x > 1$

Conclusion: The function $f(x) = 2x - x^2 + 1$ is decreasing for $x > 1$. Thus, the descending area is represented by the interval $1 < x$.

The correct choice that matches this interval is:

$1 < x$