Y = x ^ 4 + 4x [-2;1].

To analyze the function \( y = x^4 + 4x \) over the interval \([-2, 1]\), we can follow these steps: 1. **Evaluate the function at the endpoints of the interval:** - For \( x = -2 \): \[ y(-2) = (-2)^4 + 4(-2) = 16 - 8 = 8 \] - For \( x = 1 \): \[ y(1) = (1)^4 + 4(1) = 1 + 4 = 5 \] 2. **Find the critical points by taking the derivative:** - The derivative of \( y \) is: \[ y' = 4x^3 + 4 \] - Set the derivative equal to zero to find critical points: \[ 4x^3 + 4 = 0 \implies x^3 = -1 \implies x = -1 \] 3. **Evaluate \( y \) at the critical point:** - For \( x = -1 \): \[ y(-1) = (-1)^4 + 4(-1) = 1 - 4 = -3 \] 4. **Summarize the values:** - \( y(-2) = 8 \) - \( y(-1) = -3 \) - \( y(1) = 5 \) 5. **Determine the minimum and maximum values:** - The function increases from \( y(-2) = 8 \) to \( y(-1) = -3 \), then increases again to \( y(1) = 5 \). - Thus, the **minimum value** of \( y \) on the interval \([-2, 1]\) is \(-3\) at \( x = -1 \), and the **maximum value** is \( 8 \) at \( x = -2 \). In conclusion: - Minimum value: \( y = -3 \) at \( x = -1 \) - Maximum value: \( y = 8 \) at \( x = -2 \)