Find Decreasing Intervals for y = 2x^2 - 4x + 5: Quadratic Function Analysis

To determine where the function $y = 2x^2 - 4x + 5$ is decreasing, we follow these steps:

1. Calculate the Derivative:

The derivative of $y$ with respect to $x$ is given by:

$y' = \frac{d}{dx}(2x^2 - 4x + 5) = 4x - 4$

2. Find Critical Points:

Set the derivative equal to zero and solve for $x$:

$4x - 4 = 0$

$4x = 4$

$x = 1$

3. Analyze the Sign of the Derivative:

To determine where the function is decreasing, examine the sign of $y' = 4x - 4$ around $x = 1$:

• For $x < 1$, choose a test point, such as $x = 0$. Substitute into the derivative:

• $y' = 4(0) - 4 = -4$ (negative), indicating that the function is decreasing.

• For $x > 1$, choose a test point, such as $x = 2$. Substitute into the derivative:

• $y' = 4(2) - 4 = 4$ (positive), indicating that the function is increasing.

Conclusion: The function is decreasing when $x < 1$.

Therefore, the interval where the function $y = 2x^2 - 4x + 5$ is decreasing is:

$x < 1$