Question

For each of the following, determine if the given polynomial must have a global minimum, global maximum, or neither. Explain your reasoning. 15. f(x)=x⁴-5x³+x+6 16. y=-2x³-x²+8x 17. g(x)=-x⁶+x³+4x²+1

Answer 1

To determine if a polynomial has a global minimum, maximum, or neither, we need to analyze its behavior. For each given polynomial, we find the critical points by taking the derivative and setting it equal to zero. By evaluating the second derivative at these points, we can determine if they correspond to a minimum or maximum.

Step-by-step explanation:

To determine if a polynomial has a global minimum, maximum, or neither, we need to analyze its behavior.

For the polynomial f(x) = x⁴ - 5x³ + x + 6, we can find the critical points by taking the derivative and setting it equal to zero. By evaluating the second derivative at these points, we can determine if they correspond to a minimum or maximum. In this case, the critical points are (-1, 2) and (2, -12). The second derivative is positive at (-1, 2), indicating a local minimum, while it is negative at (2, -12), indicating a local maximum. Therefore, the polynomial has both a global minimum and a global maximum.

For the polynomial y = -2x³ - x² + 8x, we can follow the same process. The critical points are (-1, 5) and (0, 0). The second derivative is negative at both of these points, indicating that they correspond to local maximums. Therefore, this polynomial has two local maximums, but no global maximum.

For the polynomial g(x) = -x⁶ + x³ + 4x² + 1, again we find the critical points by taking the derivative and setting it equal to zero. The critical point is (-1, 4). Evaluating the second derivative, we find that it is positive at (-1, 4), indicating a local minimum. Therefore, this polynomial has a global minimum.