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 \( x = -2 \)