Y=x^3-12x+11

To analyze the function \( y = x^3 - 12x + 11 \), we can look for critical points, determine the behavior of the function, and analyze its graph.

1. Finding Critical Points

To find the critical points, we first need to compute the first derivative:

\[ y' = \frac{dy}{dx} = 3x^2 - 12 \]

Next, we set the first derivative equal to zero to find critical points:

\[ 3x^2 - 12 = 0 \]

Solving for \( x \):

\[ 3x^2 = 12 \\ x^2 = 4 \\ x = 2 \quad \text{or} \quad x = -2 \]

Now, we have two critical points: \( x = -2 \) and \( x = 2 \).

2. Finding the Second Derivative

Next, we find the second derivative to determine the concavity of the function:

\[ y'' = \frac{d^2y}{dx^2} = 6x \]

Now, we can test the concavity at the critical points:

• For \( x = -2 \): \[ y''(-2) = 6(-2) = -12 \quad (\text{concave down, local maximum}) \]
• For \( x = 2 \): \[ y''(2) = 6(2) = 12 \quad (\text{concave up, local minimum}) \]

3. Evaluating the Function at Critical Points

Now, let’s evaluate \( y \) at the critical points to find the corresponding \( y \)-values:

• For \( x = -2 \): \[ y(-2) = (-2)^3 - 12(-2) + 11 = -8 + 24 + 11 = 27 \]

• For \( x = 2 \): \[ y(2) = (2)^3 - 12(2) + 11 = 8 - 24 + 11 = -5 \]

4. Summary of Points

• Local Maximum at \( (-2, 27) \)
• Local Minimum at \( (2, -5) \)

5. End Behavior

Since this is a cubic function and the leading coefficient (from \( x^3 \)) is positive, the end behavior will be:

• As \( x \to -\infty, y \to -\infty \)
• As \( x \to \infty, y \to \infty \)

6. Graph Behavior

The function will rise to the left, reach a local maximum at \( (-2, 27) \), then fall to a local minimum at \( (2, -5) \), and rise again to the right.

7. Finding the Roots

To find the roots of the equation \( x^3 - 12x + 11 = 0 \), we can look for rational roots using the Rational Root Theorem or numerical methods, but since this function is cubic, it's likely it has only three real roots. A numerical method (like Newton's method) or graphing calculator would be useful here.

Conclusion

You'll want to analyze this function using the steps outlined, focusing on critical points, concavity, and end behavior to understand the characteristics of the curve. You might be asked to sketch the graph or solve for roots analytically or numerically in an exam setting.