Question
45. The function h is definied as h(x)=(x+5)^2(x-2) The cubic function f(x) has a root at -3 and double root at 4, and a leading coefficient of 1. What is the horizontal distance, in units between the relative maximum of h (x) and the relative minimum of f(x) 46. What is the diameter of the circle that is defined by x^2+y^2+8x-2y-104=0
Solution
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Resposta
45. To find the horizontal distance between the relative maximum of h(x) and the relative minimum of f(x), we need to first find the x-coordinates of the relative maximum of h(x) and the relative minimum of f(x).The function h(x) is a cubic function, and its graph will have a relative maximum and a relative minimum. To find the x-coordinate of the relative maximum, we need to find the derivative of h(x) and set it equal to zero to find the critical points. The derivative of h(x) is:h'(x) = 2(x+5)(x-2) + (x+5)^2Setting h'(x) equal to zero and solving for x, we get:2(x+5)(x-2) + (x+5)^2 = 02(x+5)(x-2) = -(x+5)^22(x-2) = -(x+5)x = -7So the x-coordinate of the relative maximum of h(x) is -7.The cubic function f(x) has a root at -3 and a double root at 4, and a leading coefficient of 1. This means that the function f(x) can be written as:f(x) = (x+3)(x-4)^2The relative minimum of f(x) occurs at the double root, which is x = 4.Therefore, the horizontal distance between the relative maximum of h(x) and the relative minimum of f(x) is |-7 - 4| = 11 units.46. To find the diameter of the circle defined by the equation x^2 + y^2 + 8x - 2y - 104 = 0, we need to rewrite the equation in the standard form of a circle equation, which is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.To do this, we need to complete the square for the x and y terms. Completing the square for the x terms, we get:x^2 + 8x = (x+4)^2 - 16Completing the square for the y terms, we get:y^2 - 2y = (y-1)^2 - 1Substituting these into the original equation, we get:(x+4)^2 - 16 + (y-1)^2 - 1 = 104(x+4)^2 + (y-1)^2 = 121Comparing this to the standard form of a circle equation, we can see that the center of the circle is (-4,1) and the radius is the square root of 121, which is 11. Therefore, the diameter of the circle is 2 * 11 = 22 units.