answer:To solve, the negation of the proposition "If x geq 1, then x^2 - 4x + 2 geq -1" is: If x < 1, then x^2 - 4x + 2 < -1; Therefore, the answer is: If x < 1, then x^2 - 4x + 2 < -1. This is directly derived by using the contrapositive relationship among the four types of propositions. This question examines the application of the contrapositive relationship among the four types of propositions. It is important to distinguish between the negation of a proposition and its negation proposition, making it a basic question. Thus, the final answer is boxed{text{If } x < 1, text{ then } x^2 - 4x + 2 < -1}.

answer:The perimeter of a hexagon is the sum of the lengths of all its sides. Since a hexagon has six sides and each side is 4 inches long, the perimeter is: Perimeter = 6 sides × 4 inches/side = 24 inches So, the perimeter of the hexagon is boxed{24} inches.

answer:Given, we need to evaluate log_4 16. We start by expressing 16 as a power of 4: [ 16 = 4^2 ] Thus, using the definition of logarithms, [ log_4 16 = log_4 (4^2) ] Using the power rule of logarithms, which states log_b(a^c) = c log_b a, we get: [ log_4 (4^2) = 2 log_4 4 ] Since log_4 4 = 1 (because 4^1 = 4), [ 2 log_4 4 = 2 times 1 = 2 ] Thus, we have: [ log_4 16 = boxed{2} ]

answer:In spherical coordinates (rho, theta, phi): - rho denotes the radial distance from the origin. - phi is the polar angle measured from the positive z-axis. The given equation is rho = c sin phi. This expression suggests that the radius rho varies with the sine of the polar angle phi. This results in a surface generated by a circle of radius c revolved around the z-axis, known as a cone with its vertex at the origin. We consider a vertical cross-section of this object. When rho = c sin phi, as phi changes from 0 (where sin phi = 0 and rho = 0) to pi/2 (where sin phi = 1 and rho = c), the form of the surface is a right circular cone expanding outward from the origin up to rho = c at phi = pi/2, then tapers back to a point at phi = pi. Hence, the answer is boxed{text{(F)}} corresponding to a cone.