answer:Given tanalpha=2, then sin2alpha= frac {2sinalphacosalpha}{sin^{2}alpha+cos^{2}alpha}=frac {2tanalpha}{tan^{2}alpha+1}=frac {4}{4+1}=frac {4}{5}, Therefore, the correct option is: boxed{D}. This problem involves using the basic relationship between trigonometric functions of the same angle and the double-angle formula to find the value of sin2alpha. It mainly tests the application of the basic relationship between trigonometric functions of the same angle and the double-angle formula, and is considered a basic question.

answer:Given that the random variable xi follows a normal distribution with mean Exi = 3 and variance Dxi = 4, we are asked to find E(2xi + 1) and D(2xi + 1). For E(2xi + 1), we use the linearity of expectation which states that E(axi + b) = aExi + b for any constants a and b. Substituting a = 2 and b = 1, we get: [E(2xi + 1) = 2Exi + 1 = 2 times 3 + 1 = 6 + 1 = 7.] For D(2xi + 1), we use the property of variance which states that D(axi + b) = a^2Dxi for any constants a and b. Note that the addition of a constant (+1 in this case) does not affect the variance. Substituting a = 2, we find: [D(2xi + 1) = 4Dxi = 4 times 4 = 16.] Therefore, the values of E(2xi +1) and D(2xi +1) are respectively 7 and 16. This matches with option boxed{text{D}}.

answer:Solution: f(-x)= frac {4^{-x}+1}{2^{-x}}= frac {1+4^{x}}{2^{x}}=f(x), Therefore, f(x) is an even function, and its graph is symmetric about the y-axis. Hence, the correct choice is boxed{text{D}}. The intention behind the given condition is not obvious. The method to solve this problem should break through from the options. Since two of the options are related to the parity (odd or even nature) of the function, it is better to first verify the parity. When examining the symmetry of a function, it is appropriate to start by studying its parity.

answer:1. Consider the given expressions: [ (a+b+c)^{2} - 8ac, quad (a+b+c)^{2} - 8bc, quad (a+b+c)^{2} - 8ab ] 2. To proceed, we first sum all three of these expressions: [ left((a+b+c)^{2} - 8acright) + left((a+b+c)^{2} - 8bcright) + left((a+b+c)^{2} - 8abright) ] 3. Expand and combine the expressions: [ (a+b+c)^{2} + (a+b+c)^{2} + (a+b+c)^{2} - 8ac - 8bc - 8ab ] [ = 3(a+b+c)^{2} - 8ac - 8bc - 8ab ] 4. Write ((a+b+c)^{2}) in expanded form: [ (a+b+c)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2ac + 2bc ] Thus: [ 3(a+b+c)^{2} = 3(a^{2} + b^{2} + c^{2} + 2ab + 2ac + 2bc) ] 5. Substitute this expanded form back into our combined expressions: [ 3(a^{2} + b^{2} + c^{2} + 2ab + 2ac + 2bc) - 8ac - 8bc - 8ab ] 6. Distribute and combine like terms: [ 3a^{2} + 3b^{2} + 3c^{2} + 6ab + 6ac + 6bc - 8ac - 8bc - 8ab ] [ = 3a^{2} + 3b^{2} + 3c^{2} + 6ab + 6ac + 6bc - 8ac - 8bc - 8ab ] Combine like terms again: [ = 3a^{2} + 3b^{2} + 3c^{2} - 2ab - 2ac - 2bc ] 7. Rearrange the expression: [ = a^{2} + b^{2} + c^{2} + a^{2} + b^{2} + c^{2} + a^{2} + b^{2} + c^{2} - 2ab - 2ac - 2bc ] [ = a^{2} + b^{2} + c^{2} + (a-b)^{2} + (a-c)^{2} + (b-c)^{2} ] 8. Since (a), (b), and (c) are positive, each of the squared terms ((a-b)^{2}), ((a-c)^{2}), and ((b-c)^{2}) are non-negative. 9. Hence, their sum (a^2 + b^2 + c^2 + (a-b)^2 + (a-c)^2 + (b-c)^2) is positive. 10. Given that the total sum of our expressions is positive and since it is a sum of three terms, at least one of these terms must be positive. # Conclusion: Therefore, we have proved that at least one of the expressions ((a+b+c)^{2} - 8ac), ((a+b+c)^{2} - 8bc), or ((a+b+c)^{2} - 8ab) is positive. [ blacksquare ]