Optimising a Road Tunnels Gradient?

📌 CONCEPT: The problem requires finding the minimum value of a function that represents the tunnel's gradient, which can be achieved by finding the critical points of the function and determining the nature of these points.

📐 RULE / FORMULA: To find the critical points, we need to find the derivative of the given function and set it equal to zero, then solve for x. The formula for the derivative of the function y = - (x^2 + 2x + 1) / 3 is y' = - (2x + 2) / 3.

💡 WORKED EXAMPLE: Let's find the critical points of the function y = - (x^2 + 2x + 1) / 3. First, we find the derivative: y' = - (2x + 2) / 3. Setting y' = 0, we get - (2x + 2) / 3 = 0. Solving for x, we get x = -1. To determine the nature of this point, we can use the second derivative test. The second derivative of the function is y'' = -2/3. Since y'' < 0, x = -1 is a point of local maximum, and the minimum value of the function occurs at x = -1.

⚠️ COMMON MISTAKE: Students often forget to check the nature of the critical points, which can lead to incorrect conclusions about the minimum or maximum value of the function.