Math topic中文

Increasing decreasing

increasing and decreasing functions

Practice questions

  1. 1A curve has the equation y=2x−33x+4y = \frac{2x - 3}{3x + 4}, where x≠−43x \ne -\frac{4}{3}. The normal to the point PP on the cur…
  2. 2Given that y=6x22−5xy = \frac{6x^2}{2-5x}, find the range of values of xx for which yy is a decreasing function of xx.
  3. 3已知函数f(x)=x2−ax(a≠0)f(x)=x^{2}-ax(a\neq 0)，g(x)=ln⁡xg(x)=\ln x，f(x)f(x)图象与xx轴异于原点的交点MM处的切线为l1l_{1}，g(x−1)g(x-1)与xx轴的交点NN处的切线为…
  4. 4已知函数f(x)=ex−1f(x)={{e}^{x}}-1，g(x)=sin⁡x(1)g(x)=\sin x(1)判断F(x)=f(x)−g(x)F(x)=f(x)-g(x)在x∈[0,+∞)x\in[0,+\infty)上零点的个数； (2)当x∈[0,π]x\in[0,\pi]时，f(x)⩾ag(x)(a∈R)f(x)\geqslant ag(x)(a\in R)…
  5. 5The equation of a curve is given by y=5xe2x+1y = 5xe^{2x+1}. Find dydx\frac{dy}{dx}. Hence, find ∫xe2x+1 dx\int xe^{2x+1} \text{ d}x. Det…
  6. 6A curve has the equation y=e2x+1x+3y = \frac{e^{2x+1}}{x+3}, where x≠−3x \ne -3. Find an expression for dydx\frac{\mathrm{d}y}{\mathrm{d}x}…
  7. 7The equation of a curve is y=e−2xtan⁡xy = e^{-2x}\tan x. Show that dydx=e−2x(1−tan⁡x)2\dfrac{dy}{dx} = e^{-2x}(1 - \tan x)^2. Find, in terms of π\pi…
  8. 8A curve has the equation y=x3+kx2+kx+8y = x^3 + kx^2 + kx + 8. Find the set of values of kk such that the curve is an increasing f…
  9. 9已知等差数列{an}\{a_{n}\}的前nn项和为Sn(n∈N∗)S_{n}(n∈N^{*})，且an=2n+λa_{n}=2n+λ，若数列{Sn}\{S_{n}\}在n⩾7n\geqslant 7时为递增数列，则实数λλ的取值范围为()()…
  10. 10Find the range of values of xx such that the graph y=ln⁡(x−2x−3)2y = \ln \left( \frac{x-2}{x-3} \right)^2 is decreasing.
  11. 11已知 a>0a > 0，函数 f(x)=x  ⁣2+aln⁡x−axf(x)=x{\,\!}^{2}+a\ln x- ax 在(0,+∞)(0,+∞)上是增函数，则 aa 的范围为__
  12. 12已知函数g(x)=1x⋅sin⁡θ+ln⁡x在[1,+∞)g(x)= \dfrac {1}{x\cdot \sin \theta }+\ln x{在}[1,+∞)上为增函数，且θ∈(0,π),f(x)=mx−m−1+2ex−ln⁡x,m∈Rθ∈(0,π),f(x)=mx- \dfrac {m-1+2e}{x}-\ln x,m∈R． (1…
  13. 13It is given that y=x3+px2+qx+10y = x^3 + px^2 + qx + 10 where pp and qq are integers. The only values of xx for which yy is a de…
  14. 14函数f(x)=ax3+(a−1)x2+48(a−2)x+bf(x)=ax^{3}+(a-1)x^{2}+48(a-2)x+b的图象关于原点对称，则f(x)f(x)在[−4,4][-4,4]上()() A. 单调递增 B. 单调递减 C. [−4,0][-4,0]单调递增，[0,4][0,4]…
  15. 15Given that y=16x1+x2y = \frac{16x}{1+x^2}, find the range of values of xx for which yy is increasing. Another curve has equat…
  16. 16已知幂函数f(x)=(−m2+2m+4)xmf(x)=(-m^{2}+2m+4)x^{m}在(0,+∞)(0,+∞)上单调递减，则实数m=m=___.
  17. 17The function is defined by g(x)=x2−4x2+6g(x) = \frac{x^2 - 4}{x^2 + 6}, x>0x > 0. Explain, with working, whether g(x)g(x) is an increa…
  18. 18Find the range of xx such that y=3x2e5−2xy = 3x^2 e^{5-2x} is a decreasing function.
  19. 19The function ff is defined, for all values of xx, by f(x)=(x+3)(1−2x)2f(x)=(x+3)(1-2x)^2. Find the range of values of xx for which ff…
  20. 20Answer ALL the questions The function ff is defined, for all values of xx, by f(x)=(2x−x2)exf(x) = (2x - x^2)e^x. Find the ran…
  21. 21Find ddx(ln⁡x4x).Hence,determinetherangeofvaluesof\dfrac{d}{dx}\left(\dfrac{\ln x}{4x}\right). Hence, determine the range of values of xforwhichthecurvefor which the curvey=\d…
  22. 22设数列{an}\{a_{n}\}满足an+1=4an−2an+1(n∈N∗)(1)a_{n+1}= \dfrac {4a_{n}-2}{a_{n}+1}(n∈N^{*})(1)若a1=3a_{1}=3，bn=2−anan−1(n∈N∗)b_{n}= \dfrac {2-a_{n}}{a_{n}-1}(n∈N^{*})…
  23. 23设函数f(x)=ln⁡xf(x)=\ln x，g(x)=xx+1g(x)=\dfrac{x}{x+1}，记F(x)=f(x)−g(x)(1)F(x)=f(x)-g(x)(1)求曲线y=f(x)y=f(x)在x=ex=e处的切线方程； (2)求函数F(x)F(x)在[1e,e2][\dfrac{1}{e},e^{2}]…
  24. 24已知函数f(x)=−x4+8x2+2021f(x)=-x^{4}+8x^{2}+2021，则f(x)f(x)的一个单调增区间为()(\quad) A. (−3,0)(-3,0) B. (0,1)(0,1) C. (1,3)(1,3) D. (2,4)(2,4)
  25. 25Given that y=x22x−1y = x^2 \sqrt{2x-1}, show that dydx=x(5x−2)2x−1\frac{\text{d}y}{\text{d}x} = \frac{x(5x-2)}{\sqrt{2x-1}}. Determine wheth…
  26. 26A particle moves along the curve y=23x+1y = \frac{2}{\sqrt{3x+1}} in such a way that the yy-coordinate of the particle is de…
  27. 27已知函数f(x)=axexf(x)= \dfrac {ax}{e^{x}}在x=0x=0处的切线方程为y=xy=x． (1)求aa的值； (2)若对任意的x∈(0,2)x∈(0,2)，都有1f(x)>k+2x−x2 \dfrac {1}{f(x)} > k+2x-x^{2}…
  28. 28已知y=f(x)y=f(x)是定义在RR上的函数，且f(1)=1f(1)=1，f′(x)>1f′(x) > 1，则f(x)>xf(x) > x的解集是()() A. (0,1)(0,1) B. (−1,0)∪(0,1)(-1,0)∪(0,1) C. (1,+∞)(1,+∞)…
  29. 29设锐角△ABC\triangle ABC的三个内角AA，BB，CC的对边分别为aa，bb，cc，且c=1c=1，A=2CA=2C，则△ABC\triangle ABC周长的取值范围为()(\quad)…
  30. 30已知函数f(x)=mln⁡xf(x)=m\ln x． (1)讨论函数F(x)=f(x)+1x−1F(x)=f(x)+ \dfrac {1}{x}-1的单调性； (2)定义：“对于在区域DD上有定义的函数y=f(x)y=f(x)和y=g(x)y=g(x)，若满足f(x)⩽g(x)f(x)\leqslant g(x)…
  31. 31设函数f(x)=ax−ax−2ln⁡xf(x)=ax- \dfrac {a}{x}-2\ln x． (1)若f(x)f(x)在x=2x=2时有极值，求实数aa的值和f(x)f(x)的极大值； (2)若f(x)f(x)在定义域上是增函数，求实数aa的取…
  32. 32The equation of a curve is y=x2(6−x)y = x^2(6 - x). Find the coordinates of the stationary points of the curve. Use the second…
  33. 33当n<0n < 0时，幂函数y=xny=x^{n}是定义域上的减函数.().() A. √√ B. ××
  34. 34The function f is defined, for x>32x > \frac{3}{2}, by f(x)=e2x1−x\text{f}(x) = \frac{e^{2x}}{1 - x}. Explain clearly whether f is…
  35. 35Given that y=2x2x−3y = \frac{2x^2}{x - 3}, determine the range of values of xx for which yy is a decreasing function.
  36. 36It is given that y=2x−1x+2y = \frac{2x-1}{\sqrt{x+2}} for x>qx > q. State the value of qq. Find dydx\dfrac{dy}{dx}. State whether…
  37. 37Given that y=ln⁡(x+3)33x+9y = \frac{\ln(x+3)^3}{3x+9} for x>−3x > -3, find the set of values of xx for which yy is an increasing func…
  38. 38Answer all questions. Differentiate 2xe1−2x2xe^{1-2x} with respect to xx. Hence find the exact value of ∫012xe1−2xdx\int_{0}^{1} 2xe^{1-2x} \text{d}x…
  39. 39The equation of a curve is y=e2x1−xy = \frac{e^{2x}}{1-x}, where x>32x > \frac{3}{2}. Find dydx\frac{dy}{dx} and explain whether yy…
  40. 40已知n∈N∗n\in N^{*}，集合Mn={12,34,58,M_{n}=\{\dfrac{1}{2},\dfrac{3}{4},\dfrac{5}{8},…2n−12n}\dfrac{2n-1}{2^{n}}\}，集合MnM_{n}所有非空子集的最小元素之…
  41. 41The function ff is defined by f(x)=3sin⁡x−4cos⁡2xf(x)=3\sin x-4\cos 2x, 0<x<π20<x<\dfrac{\pi}{2}. Explain with working, whether ff is an in…
  42. 42从A、B两水库向甲、乙两地调水，其中甲地需水15万吨，乙地需水13万吨，A、B两水库各可调出水14万吨．从A地到甲地50千米，到乙地30千米；从B地到甲地60千米，到乙地45千米．设计一个调运方案使水的调运量（单位：万吨•千米）尽可能小．
  43. 43若直线y=kx（k≠0）经过点（-2，6），则y随x的增大而 ___ ．
  44. 44若函数f(x)=x2+2(a−1)x+2f(x)=x^{2}+2(a-1)x+2在区间(−∞,3)(-∞,3)上是单调函数，则实数aa的取值范围是()(\quad) A. (−∞,−2](-∞,-2] B. [−2,+∞)[-2,+∞) C. (−∞,4](-∞,4] D.…
  45. 45已知函数f(x)=−2xln⁡x+x2−2ax+a2.f(x)=-2x\ln x+x^{2}-2ax+a^{2}.记g(x)g(x)为f(x)f(x)的导函数． (1)若曲线y=f(x)y=f(x)在点(1,f(1))(1,f(1))处的切线垂直于直线x+y+3=0x+y+3=0，求aa的…
  46. 46已知幂函数f(x)=(m−1)2xm2−4m+2f(x){=}{(m{-}1)}^{2}x^{m^{2}{-}4m{+}2}在(0,+∞)(0{,}{+∞})上单调递增，函数g(x)=2x−tg(x){=}2^{x}{-}t，对于任意的x1∈[1,6)x_{1}{∈[}1{,}6)，总存在x2∈[1,6)x_{2}{∈[}1{,}6)…
  47. 47已知函数f(x)=ax2−bln⁡xf(x)=ax^{2}-b\ln x在点A(1,f(1))A(1,f(1))处的切线方程为y=1y=1； (1)求实数aa，bb的值； (2)求函数f(x)f(x)的极值．
  48. 48Find the equation of the normal to the curve y=2x−6x−2y = \frac{2x-6}{x-2} at the point where the curve meets the y-axis. Deter…
  49. 49已知函数 F(x)=px+ln⁡⁡(px)(F(x)= \dfrac{p}{x}+\ln ⁡(px) (其中 p>0).(1)p > 0).(1)当 p=12p=\dfrac{1}{2} 时，判断 F(x)F(x) 零点的个数kk； (2)在(1)的条件下，记这些零点…
  50. 50已知函数f(x)=13x3−(2k+1)x2+3k(k+2)x+1f(x)= \dfrac {1}{3}x^{3}-(2k+1)x^{2}+3k(k+2)x+1，其中kk为实数． (1)当k=−1k=-1时，求函数f(x)f(x)在[0,6][0,6]上的最大值和最小值； (2)求函数…
  51. 51已知函数f(x)=sin⁡2x+ax2.(1)f(x)=\sin^{2}x+ax^{2}.(1)当a=1a=1时，求f(x)f(x)的单调区间； (2)若x∈[0,π2]x\in\left[0,\dfrac{\pi}{2}\right]，不等式sin⁡(2cos⁡ x)+a2x2⩾af(x)\sin(2\cos\:x)+a^{2}x^{2}\geqslant af(x)…
  52. 52已知数列{an}\{a_{n}\}满足a1=1a_{1}\text{=}1，an+1=an2+3an(n∈N*)a_{n\text{+}1}\text{=}\dfrac{a_{n}}{2\text{+}3a_{n}}(n{\in}N^{\text{*}})，则下列结论…
  53. 53The diagrams AA, BB, CC, DD, EE and FF show six graphs of different functions. Select the diagram that best shows…
  54. 54A curve is such that d2ydx2=−4sin⁡(2x+π)\dfrac{\mathrm{d}^2 y}{\mathrm{d}x^2} = -4\sin(2x+\pi) and the point P(π2,π)P\left(\dfrac{\pi}{2},\pi\right)…
  55. 55It is given that y=(x−2)(2x−5)3y = (x - 2)(2x - 5)^3. Obtain an expression for dydx\frac{dy}{dx} in the form (ax+b)(2x−5)2(ax + b)(2x - 5)^2, whe…