Final answer:
To determine the global minimum or maximum, we analyze the degree and the leading coefficient of the polynomial. Quadratic polynomials have either a global minimum or maximum depending on the sign of the leading coefficient. Polynomials with odd degrees and a non-zero leading coefficient extend infinitely in opposite directions, having neither a global minimum nor maximum.
Step-by-step explanation:
When determining if a given polynomial must have a global minimum, global maximum, or neither, we assess the degree and leading coefficient of the polynomial.
a) f(x) = x² + 3x + 5: This is a quadratic polynomial with a positive leading coefficient. Such polynomials open upwards and therefore always have a global minimum but no global maximum.
b) g(x) = -2x³ + 6x² - 4x + 1: The leading term has a negative coefficient and an odd degree, indicating that the ends of the graph extend indefinitely in opposite directions. Hence, this polynomial has neither a global minimum nor maximum.
c) h(x) = 4x⁴ - 8x³ + 2x² - 6x + 9: This is a quartic polynomial with a positive leading coefficient. Since all even degree polynomials with positive leading coefficients tend towards positive infinity as x approaches both positive and negative infinity, this polynomial must have a global minimum but no global maximum.
d) k(x) = -x² + 4x - 2: As with the first example, this is a quadratic polynomial; however, the leading coefficient is negative, indicating the parabola opens downwards. Therefore, it has a global maximum and no global minimum.