Asosiy tarkibga o'tish

4.6-bo'lim : Grafik shakli, II qism

1 va 2 masalalar uchun funksiya grafigi berilgan. Funktsiya yuqoriga va pastga botiq bo'lgan intervallarni aniqlang.

  1. This graph has no y scale. It starts in the third quadrant at x=-4 and increases to a peak at approximately x=-2.8 in the second quadrant and then starts to decrease and the graph is cupped downwards as it does this. At x=-2 the graph is still decreasing but now switches over to cupped vaguely upwards. This behavior continues until it reaches the origin where it is now decreasing with a vaguely cupped downward behavior. At x=2 the graph continues to decrease but switches over to being cupped upwards. The graph hits a valley at approximately x=2.8 in the fourth quadrant and then increases until it ends at x=4 in the first quadrant and the graph is still cupped upwards.
  2. This graph has no y scale. It starts in the second quadrant at x=-6 and increases to a peak at approximately x=-5.2 and then starts to decrease and the graph is cupped downwards as it does this. At x=-4 the graph is still decreasing but now switches over to cupped upwards. This behavior continues until it reaches a valley in the third quadrant at x=-2. At this point the graph now starts to increase and remains cupped upward. At x=-1 the graph is still increasing but switches over to being cupped downwards. The graph hits a peak at the origin and starts to decrease while still being cupped downward. At x=1 the graph is still decreasing but switches over to being cupped upwards. The graph hits a valley at approximately x=1.8 in the fourth quadrant and then starts to increase and continues to be cupped upward. This behavior continues until the graph ends at x=3 in the first quadrant.

3-5-masalalar uchun grafigi 2nd hosila funktsiya berilgan. Ushbu grafikdan qaysi intervallarni aniqlang funktsiyasi yuqoriga va pastga botiq bo'ladi.

  1. This graph has no y scale. The graph starts at x=-4 in the second quadrant and goes down through the x-axis at x=-3 and hits a valley at approximately x=-1.5. It now goes up through the x-axis at x=1 and hits a peak at approximately x=3.2. It then goes back down through the x-axis at x=5 and ends at x=6 in the fourth quadrant.
  2. This graph has no y scale. The graph starts at x=-2 in the second quadrant and goes down and just touches the x-axis at x=-1 and then hits a peak at x=2. It now goes down through the x-axis at x=4 and hits a valley in the fourth quadrant at approximately x=5.8. It then goes back up through the x-axis at x=7 and ends at x=8 in the first quadrant.
  3. This graph has no y scale. The graph starts at x=-2 in the second quadrant and goes down to a peak, still in the second quadrant at approximately x=-0.5. It then goes up to a peak in the first quadrant at approximately x=1.5 and then goes down and just touches the x-axis at x=4. The graph then goes up to finish at around x=6 in the first quadrant.

6-18 muammolar uchun quyidagi savollarga javob bering.

  1. Funktsiya yuqoriga va pastga botiq bo'lgan intervallarni aniqlang.
  2. Funksiyaning burilish nuqtalarini aniqlang.


  1. \(f\left( x \right) = {x^3} + 9{x^2} + 24x - 6\)
  2. \(Q\left( t \right) = {t^4} - 2{t^3} - 120{t^2} - 84t + 35\)
  3. \(h\left( z \right) = 3{z^5} - 20{z^4} + 40{z^3}\)
  4. \(g\left( w \right) = 5{w^4} - 2{w^3} - 18{w^2} + 108w - 12\)
  5. \(g\left( x \right) = 10 + 360x + 20{x^4} + 3{x^5} - {x^6}\)
  6. \(\left[ { - 20,11} \right]\) da \(A\left( x \right) = 9x - 3{x^2} - 160\sin \left( {{\displaystyle \frac{x}{4}}} \right)\)
  7. \(f\left( x \right) = 3\cos \left( {2x} \right) - {x^2} - 14\)on \(\left[ {0,6} \right]\)
  8. \(h\left( t \right) = 1 + 2{t^2} - \sin \left( {2t} \right)\) \(\left[ { - 2,4} \right]\) da
  9. \(R\left( v \right) = v{\left( {v - 8} \right)^{{\frac{1}{3}}}}\)
  10. \(g\left( x \right) = \left( {x - 1} \right){\left( {x + 3} \right)^{{\frac{2}{5}}}}\)
  11. \(f\left( x \right) = {{\bf{e}}^{4\,x}} - {{\bf{e}}^{ - \,x}}\)
  12. \(h\left( w \right) = {w^2}{{\bf{e}}^{ - w}}\)
  13. \(A\left( w \right) = {w^2} - \ln \left( {{w^2} + 1} \right)\)

19-33-masalalar uchun quyidagi savollarga javob bering.

  1. Funktsiyaning kritik nuqtalarini aniqlang.
  2. Funksiyaning ortishi va kamayishi oraliqlarini aniqlang.
  3. Kritik nuqtalarni nisbiy maksimal, nisbiy minimum yoki hech biri sifatida tasniflang.
  4. Funktsiya yuqoriga va pastga botiq bo'lgan intervallarni aniqlang.
  5. Funksiyaning burilish nuqtalarini aniqlang.
  6. Bosqichlardagi ma'lumotlardan foydalaning (a) – (e) funksiyaning grafigini chizish uchun.

  1. \(f\left( x \right) = 10 - 30{x^2} + 2{x^3}\)
  2. \(G\left( t \right) = 14 + 4{t^3} - {t^4}\)
  3. \(h\left( w \right) = {w^4} + 4{w^3} - 18{w^2} - 9\)
  4. \(g\left( z \right) = 10{z^3} + 10{z^4} + 3{z^5}\)
  5. \(f\left( z \right) = {z^6} - 9{z^5} + 20{z^4} + 10\)
  6. \(Q\left( t \right) = 3t - 5\sin \left( {2t} \right)\), \(\left[ { - 1,4} \right]\)
  7. \(\left[ { - 25,0} \right]\) da \(g\left( x \right) = {\displaystyle \frac{1}{2}}x + \cos \left( {{\displaystyle \frac{1}{3}}x} \right)\)
  8. \(h\left( x \right) = x{\left( {x - 4} \right)^{{\frac{1}{3}}}}\)
  9. \(f\left( t \right) = t\,\,\sqrt {{t^2} + 1} \)
  10. \(A\left( z \right) = {z^{{\frac{4}{5}}}}\,\left( {z - 27} \right)\)
  11. \(g\left( w \right) = {{\bf{e}}^{4\,w}} - {{\bf{e}}^{6\,w}}\)
  12. \(P\left( t \right) = 3t{{\bf{e}}^{1 - {\frac{1}{4}}{t^{\,2}}}}\)
  13. \(g\left( x \right) = {\left( {x + 1} \right)^3}{{\bf{e}}^{ - \,x}}\)
  14. \(h\left( z \right) = \ln \left( {{z^2} + z + 1} \right)\)
  15. \(f\left( w \right) = 2w - 8\ln \left( {{w^2} + 4} \right)\)
  16. Quyidagi savollarning har biriga javob bering.
    1. Aniq ikkita burilish nuqtasi bo'lgan ko'phadning minimal darajasi qancha.
    2. To'liq uchta burilish nuqtasi bo'lgan ko'phadning minimal darajasi qancha.
    3. Aynan \(n\) burilish nuqtalariga ega bo'lgan ko'phadning minimal darajasi qancha.
  17. Ba'zi \(f\left( x \right)\) funksiyasi uchun \(x = 3\) da burilish nuqtasi mavjudligi ma'lum. Ushbu funktsiya haqida quyidagi savollarning har biriga javob bering.
    1. Eng oddiy shakl qaysi 2nd bu funktsiyaning hosilasi?
    2. Javobingizdan foydalanib (a) funktsiyaning o'zi olishi mumkin bo'lgan eng umumiy shaklni aniqlang.
    3. \(f\left( 0 \right) = - 6\) va \(f\left( 3 \right) = 1\) \(x = 3\) da burilish nuqtasiga ega bo'lgan funksiyani topishini hisobga olsak.

36 – 39 masalalar uchun \(f\left( x \right)\) ko'phad hisoblanadi. 2 ni hisobga olgan holdand funktsiyaning hosilasi, agar iloji bo'lsa, berilgan kritik nuqtalarning har birini nisbiy minimal yoki nisbiy maksimal deb tasniflang. Kritik nuqta(lar)ni tasniflashning iloji bo'lmasa, nima uchun ularni tasniflash mumkin emasligini aniq tushuntiring.

  1. \(f''\left( x \right) = 3{x^2} - 4x - 15\). Kritik nuqtalar: \(x = - 3\), \(x = 0\) va \(x = 5\).
  2. \(f''\left( x \right) = 4{x^3} - 21{x^2} - 24x + 68\). Kritik nuqtalar: \(x = - 2\), \(x = 4\) va \(x = 7\).
  3. \(f''\left( x \right) = 23 + 18x - 9{x^2} - 4{x^3}\). Kritik nuqtalar: \(x = - 4\), \(x = - 1\) va \(x = 3\).
  4. \(f''\left( x \right) = 216 - 410x + 249{x^2} - 60{x^3} + 5{x^4}\). Kritik nuqtalar: \(x = 1\), \(x = 4\) va \(x = 5\).
  5. Ushbu muammo uchun \(f\left( x \right) = {\left( {x + 1} \right)^3}{\left( {x - 1} \right)^4}\) dan foydalaning.
    1. Funktsiya uchun kritik nuqtalarni aniqlang.
    2. 2 dan foydalaningnd kritik nuqtalarni nisbiy minimal yoki nisbiy maksimal deb tasniflash uchun hosila testi. Kritik nuqta(lar)ni tasniflashning iloji bo'lmasa, nima uchun ularni tasniflash mumkin emasligini aniq tushuntiring.
    3. 1 dan foydalaningst kritik nuqtalarni nisbiy minimal, nisbiy maksimal yoki hech biri sifatida tasniflash uchun hosila testi.
  6. \(f\left( x \right)\) va \(g\left( x \right)\) konkav pastga funksiyalar ekanligini hisobga olsak. Agar biz \(h\left( x \right) = f\left( x \right) + g\left( x \right)\)ni aniqlasak, \(h\left( x \right)\) konkav pastga funktsiya ekanligini ko'rsatadi.
  7. \(f\left( x \right)\) konkav funksiya ekanligini hisobga olsak. \(g\left( x \right)\) da \(h\left( x \right) = f\left( x \right) + g\left( x \right)\) konkav funksiyasi bo'ladigan shartni aniqlang.
  8. \(f\left( x \right)\) funksiyasi uchun \(f\left( x \right)\) da \(h\left( x \right) = {\left[ {f\left( x \right)} \right]^2}\) konkav funksiya bo'ladigan shartlarni aniqlang. E'tibor bering, bu erda foydalanish mumkin bo'lgan bir nechta shartlar to'plami mavjud. Ulardan qanchasini topa olasiz?