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question:Which of the following functions is a decreasing function on the interval (0, +infty)? A: y=x+1 B: y=log_{frac{1}{2}}x C: y=2^x D: y=(x1)^2
answer:To determine which function is a decreasing function on the interval (0, +infty), let's analyze each option: For A, the function y=x+1 is a linear function. It increases continuously over the whole real number line, including the interval (0, +infty). Therefore, it is not a decreasing function on the given interval. For B, the function y=log_{frac{1}{2}}x is a logarithmic function with a base of frac{1}{2}, which is less than 1. Logarithmic functions with bases less than 1 are decreasing functions on the interval (0, +infty) since as x increases, the output decreases. This function suits the criteria for a decreasing function on the given interval. For C, the function y=2^x is an exponential function with a base greater than 1. Such functions are always increasing for all x, and hence, y=2^x is not a decreasing function on (0, +infty). For D, the function y=-(x-1)^2 is a downward opening parabola with vertex at (1, 0). In the interval (-infty, 1), the function is increasing, but on the interval (1, +infty), it is decreasing. Yet, considering our interval starts from 0, it is not strictly decreasing over the entire interval (0, +infty) since from 0 to 1, it is increasing. Thus, it does not fulfill the condition for a decreasing function on (0, +infty). Therefore, the correct option is B boxed{B}.
question:Evaluate the expression frac{3a^{2} + frac{a^{1}}{3}}{a^2} when a= 3. textbf{(A)} frac{1}{81}qquadtextbf{(B)} frac{4}{81}qquadtextbf{(C)} frac{1}{27}qquadtextbf{(D)} frac{1}{9}qquadtextbf{(E)} frac{1}{3}
answer:1. **Calculate a^{-1} and a^{-2}**: Since a^{-2} is the reciprocal of a^2 and a^{-1} is the reciprocal of a, given a = 3, we have: [ a^{-1} = left(3right)^{-1} = frac{1}{3}, quad a^{-2} = left(3right)^{-2} = left(frac{1}{3}right)^2 = frac{1}{9}. ] 2. **Substitute a^{-1} and a^{-2} into the expression**: Replace a^{-1} with frac{1}{3} and a^{-2} with frac{1}{9} in the expression: [ frac{3 cdot frac{1}{9} + frac{frac{1}{3}}{3}}{3^2} = frac{frac{1}{3} + frac{1}{9}}{9}. ] 3. **Combine the fractions in the numerator and simplify**: The numerator simplifies to: [ frac{1}{3} + frac{1}{9} = frac{3+1}{9} = frac{4}{9}. ] Thus the whole expression becomes: [ frac{frac{4}{9}}{9} = frac{4}{81}. ] 4. **Conclude with the final answer**: The final value of the expression when a = 3 is frac{4{81}}. The final answer is boxed{textbf{(B)} frac{4}{81}}
question:With the development of technology, the Chinese economy is transitioning from Made in China to Created in China. Apart from Huawei, another Chinese tech giant has emerged, which is the global monopoly in the drone market DJI. In the field of drones, DJI has more than 4600 patent applications, making it the welldeserved leader in the drone industry. Express 4600 in scientific notation as ( ). A: 4.6times 10^{3} B: 46times 10^{2} C: 4.6times 10^{2} D: 0.46times 10^{4}
answer:To express 4600 in scientific notation, we start by identifying the significant figures and their placement in relation to the decimal point to represent the number in a form that is a product of a number and a power of 10. Step 1: Identify the significant figures in 4600, which are 4 and 6. Step 2: Place the decimal after the first significant figure, which gives us 4.6. Step 3: Count the number of places the decimal has to move to return to its original position in 4600, which is 3 places to the right. Step 4: Combine the results of steps 2 and 3 to express 4600 in scientific notation, which gives us 4.6 times 10^{3}. Therefore, the correct answer, following the given options, is: boxed{A: 4.6times 10^{3}}.
question:the Cartesian coordinate system xOy, for the line l: ax+by+c=0 and points P_{1}(x_{1},y_{1}), P_{2}(x_{2},y_{2}), denote eta=(ax_{1}+by_{1}+c)(ax_{2}+by_{2}+c). If eta < 0, then it is said that points P_{1}, P_{2} are separated by the line l. If the curve C has no common points with the line l, and there exist points P_{1}, P_{2} on curve C that are separated by the line l, then the line l is called a separating line of the curve C. 1. Prove that points A(1,2), B(1,0) are separated by the line x+y1=0; 2. If the line y=kx is a separating line of the curve x^{2}4y^{2}=1, find the range of the real number k; 3. The product of the distance from the moving point M to the point Q(0,2) and its distance to the yaxis is 1. Let the trajectory of point M be curve E. Prove that among the lines passing through the origin, there is exactly one line that is a separating line of E.
answer:1. **Proof**: Substituting points (1,2) and (-1,0) into x+y-1 yields (1+2-1)(-1+0-1)=-4 < 0, therefore points (1,2) and (-1,0) are separated by the line x+y-1=0. boxed{text{Answer}} 2. **Solution**: By combining the line y=kx with the curve x^{2}-4y^{2}=1, we get (1-4k^{2})x^{2}=1. According to the problem, this equation has no solution, hence 1-4k^{2}leqslant 0, therefore kleqslant -frac{1}{2}, or kgeqslant frac{1}{2}. There are two points (-1,0) and (1,0) on the curve that are separated by the line y=kx. boxed{kleqslant -frac{1}{2} text{ or } kgeqslant frac{1}{2}} 3. **Proof**: Let point M(x,y), then sqrt{x^{2}+(y-2)^{2}}cdot|x|=1, hence the equation of curve E is [x^{2}+(y-2)^{2}]x^{2}=1. The y-axis is x=0, which obviously has no solution when combined with equation [x^{2}+(y-2)^{2}]x^{2}=1. Also, P_{1}(1,2), P_{2}(-1,2) are two points on E, and substituting into x=0, we get eta=1times(-1)=-1 < 0, hence x=0 is a separating line. If the line passing through the origin is not the y-axis, let it be y=kx, substituting into [x^{2}+(y-2)^{2}]x^{2}=1, we get [x^{2}+(kx-2)^{2}]x^{2}=1, Let f(x)=[x^{2}+(kx-2)^{2}]x^{2}-1, Since kneq 2, f(0)f(1)=-(k-2)^{2} < 0, therefore f(x)=0 has no real solution, For k=2, f(x)=[x^{2}+(2x-2)^{2}]x^{2}-1=0 has no real solution, meaning y=kx has common points with E, therefore y=kx is not a separating line of E. therefore Among the lines passing through the origin, there is exactly one line that is a separating line of E. boxed{text{Answer}}