Question

Critical points, domain endpoints, and loca -x² -5x+8,x<=1 -x²+9x-6,x>1

Answer 1

To determine the critical points, domain endpoints, and local behavior of the given quadratic equations, we need to find the derivative for each equation, set it equal to zero, and solve for x. The critical points are the solutions of the derivatives and are within the domains. The domain endpoints are x = 1 and x = positive infinity. The local behavior can be determined by analyzing the concavity of the quadratic equations at the critical points.

Step-by-step explanation:

The critical points, domain endpoints, and local behavior can be determined by analyzing the given quadratic equations:

For x ≤ 1: -x² - 5x + 8

1. Find the derivative of the equation: -2x - 5
2. Set the derivative equal to zero and solve for x: -2x - 5 = 0 => x = -2.5
3. This critical point is within the domain, so it is a critical point.

For x > 1: -x² + 9x - 6

1. Find the derivative of the equation: -2x + 9
2. Set the derivative equal to zero and solve for x: -2x + 9 = 0 => x = 4.5
3. This critical point is within the domain, so it is a critical point as well.

Therefore, the critical points for both equations are -2.5 and 4.5. The domain endpoints are x = 1 and x = positive infinity. The local behavior can be determined by analyzing the concavity of the quadratic equations at the critical points.