51. \( P(x)=x^{4}-x^{3}-10 ;[2,3] \)

To analyze the polynomial \( P(x) = x^4 - x^3 - 10 \) on the interval \([2, 3]\), we can follow these steps: 1. **Evaluate the function at the endpoints of the interval**: - Calculate \( P(2) \): \[ P(2) = 2^4 - 2^3 - 10 = 16 - 8 - 10 = -2 \] - Calculate \( P(3) \): \[ P(3) = 3^4 - 3^3 - 10 = 81 - 27 - 10 = 44 \] 2. **Check for any critical points in the interval**: - First, find the derivative \( P'(x) \): \[ P'(x) = 4x^3 - 3x^2 \] - Set the derivative equal to zero to find critical points: \[ 4x^3 - 3x^2 = 0 \] Factor out \( x^2 \): \[ x^2(4x - 3) = 0 \] This gives us \( x^2 = 0 \) (which gives \( x = 0 \), not in the interval) and \( 4x - 3 = 0 \) (which gives \( x = \frac{3}{4} \), also not in the interval). Since there are no critical points in the interval \([2, 3]\), we only need to consider the values at the endpoints. 3. **Determine the behavior of the function**: - Since \( P(2) = -2 \) and \( P(3) = 44 \), and there are no critical points in the interval, we can conclude that \( P(x) \) is continuous and increasing on \([2, 3]\). 4. **Conclusion**: - The function \( P(x) \) changes from negative to positive as \( x \) moves from 2 to 3. By the Intermediate Value Theorem, there is at least one root in the interval \((2, 3)\). Thus, we can conclude that there is at least one root of \( P(x) = 0 \) in the interval \([2, 3]\).