answer:Given the function f(x)=sin(2x+varphi), where (0 < varphi < pi), and the graph of the function is symmetric about the point (frac{2pi}{3}, 0), we proceed to analyze the given statements. # Symmetry Analysis The symmetry about the point (frac{2pi}{3}, 0) implies that for the function f(x)=sin(2x+varphi), we have: [2 times frac{2pi}{3} + varphi = kpi] [ Rightarrow varphi = kpi - frac{4pi}{3}] Given that 0 < varphi < pi, we find: [ varphi = frac{2pi}{3}] Thus, the function simplifies to: [f(x)=sin(2x+frac{2pi}{3})] # Statement A Analysis For f(x) to be monotonically decreasing, we consider the interval where frac{pi}{2} < 2x+frac{2pi}{3} < frac{3pi}{2}. Solving for x gives: [ -frac{pi}{12} < x < frac{5pi}{12} ] This interval includes (0, frac{5pi}{12}), hence f(x) is monotonically decreasing on (0, frac{5pi}{12}). Therefore, statement A is correct. # Statement B Analysis For the interval (-frac{pi}{12}, frac{11pi}{12}), we have 2x+frac{2pi}{3} in (frac{pi}{2}, frac{5pi}{2}). Given the monotonicity of f(x), there is only one critical point in this interval, making statement B incorrect. # Statement C Analysis For the axis of symmetry, setting 2x+frac{2pi}{3}=kpi + frac{pi}{2}, we solve for x and get x=frac{kpi}{2}-frac{pi}{12}, where k in mathbb{Z}. This analysis shows that statement C is incorrect. # Statement D Analysis Taking the derivative of f(x), we get: [f'(x)=2cos(2x+frac{2pi}{3})] Setting f'(x)=-1 gives cos(2x+frac{2pi}{3})=-frac{1}{2}. Solving this yields x=kpi or x=frac{pi}{3}+kpi where k in mathbb{Z}. The slope of the tangent line at (0, frac{sqrt{3}}{2}) is: [k={f'}{|}_{x=0}=2cosfrac{2pi}{3}=-1] Thus, the equation of the tangent line is y-frac{sqrt{3}}{2}=-(x-0), simplifying to y=-x+frac{sqrt{3}}{2}. Therefore, statement D is correct. # Conclusion The correct choices are A and D. Hence, the final answer is encapsulated as: [boxed{A text{ and } D}]

answer:1. Convert the logarithmic equation to its exponential form: [ 3x - 4 = 8^2 ] Simplifying the right side: [ 3x - 4 = 64 ] 2. Solve for x: [ 3x = 64 + 4 = 68 ] [ x = frac{68}{3} ] 3. Conclusion: [ boxed{x = frac{68}{3}} ]

answer:If the width of the rectangle is 6 inches and the length is 3 times the width, then the length is: 3 * 6 inches = 18 inches The area of a rectangle is found by multiplying the length by the width. So, the area of this rectangle is: Length * Width = 18 inches * 6 inches = 108 square inches Therefore, the area of the rectangle is boxed{108} square inches.

answer:Since 2 is the geometric mean of 2^{a} and 2^{b}, We have 2^{a} cdot 2^{b} = 4, This implies that a + b = 2, and frac{1}{2}(a + b) = 1, Given that a > 0 and b > 0, We can rewrite frac{1}{a} + frac{1}{b} = (frac{1}{a} + frac{1}{b})(frac{a}{2} + frac{b}{2}) = 1 + frac{b}{2a} + frac{a}{2b} geqslant 1 + 2sqrt{frac{b}{2a} cdot frac{a}{2b}} = 2, The equality holds if and only if a = b = 1. So the minimum value of frac{1}{a} + frac{1}{b} is boxed{2}. This answer corresponds to option C. The problem primarily involves the application of the basic inequality and the concept of geometric mean. By establishing the relationship between a and b through the geometric mean, we can directly find the minimum value of frac{1}{a} + frac{1}{b} using the basic inequality. This problem mainly tests the understanding of basic inequality and geometric mean, as well as computational skills.