Asosiy tarkibga o'tish

4.3-bo'lim: Minimal va maksimal qiymatlar

  1. Quyida \(f\left( x \right)\) funksiyasining grafigi keltirilgan. Funksiyaning barcha nisbiy ekstremal va mutlaq ekstremallarini aniqlang.
    The graph of \(f\left( x \right)\) on \( - 4 \le x \le 5\). Starting at approximately \(\left( { - 4, - 1} \right)\) the graph decreases to a valley at approximately \(\left( { - 3, - 1.5} \right)\), then increases to a peak at \(\left( { - 1,2} \right)\), then falls steadily to a sharp valley at \(\left( {2, - 6} \right)\), then rises steeply to a peak at \(\left( {4,5} \right)\) and finally decreases a little to approximately \(\left( {5,2} \right)\) at the right end.
    Yechim
  2. Quyida \(f\left( x \right)\) funksiyasining grafigi keltirilgan. Funksiyaning barcha nisbiy ekstremal va mutlaq ekstremallarini aniqlang.
    The graph of \(f\left( x \right)\) on \( - 5 \le x \le 9\). Starting at approximately \(\left( { - 5,6} \right)\) the graph decreases to a valley at approximately \(\left( { - 3, - 1} \right)\), rises very slightly to a small peak at approximately \(\left( { - 2, - 0.5} \right)\), then drops to another valley at approximately \(\left( { - 1, - 1} \right)\). It then rises steeply to a peak at approximately \(\left( {1,3} \right)\), drops sharply to a valley at \(\left( {2, - 4} \right)\), rises again to a peak at approximately \(\left( {6,8} \right)\) and then falls steeply, crossing the x-axis near \(x = 8\) and continuing down to approximately \(\left( {9, - 6} \right)\).
    Yechim
  3. \(g\left( x \right) = {x^2} - 4x\) grafigini chizing va quyidagi intervallarning har birida funksiyaning barcha nisbiy ekstremal va mutlaq ekstremallarini aniqlang.
    1. \(\left( { - \infty ,\infty } \right)\)
    2. \(\left[ { - 1,4} \right]\)
    3. \(\left[ {1,3} \right]\)
    4. \(\left[ {3,5} \right]\)
    5. \(\left( { - 1,5} \right]\)
    Yechim
  4. \(h\left( x \right) = - {\left( {x + 4} \right)^3}\) grafigini chizing va quyidagi oraliqlarning har birida funksiyaning barcha nisbiy ekstremal va mutlaq ekstremallarini aniqlang.
    1. \(\left( { - \infty ,\infty } \right)\)
    2. \(\left[ { - 5.5, - 2} \right]\)
    3. \(\left[ { - 4, - 3} \right)\)
    4. \(\left[ { - 4, - 3} \right]\)
    Yechim
  5. \(x = 6\)da mutlaq maksimalga va \(x = 3\)da mutlaq minimumga ega bo'lgan \(\left[ {1,6} \right]\) oralig'ida ba'zi funksiyalarning grafigini chizing. Yechim
  6. \(x = - 3\)da mutlaq maksimalga va \(x = 2\)da mutlaq minimumga ega bo‘lgan \(\left[ { - 4,3} \right]\) oralig‘idagi ba’zi funksiyalarning grafigini chizing. Yechim
  7. Quyidagi shartlarga javob beradigan funksiya grafigini chizing:
    1. Funktsiya uzluksiz.
    2. Ikki nisbiy minimumga ega.
    3. Nisbiy minimumlardan biri ham mutlaq minimum, boshqa nisbiy minimum esa mutlaq minimum emas.
    4. Bitta nisbiy maksimalga ega.
    5. Mutlaq maksimalga ega emas.
    Yechim