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4.7-bo'lim : O'rtacha qiymat teoremasi

1-4-masalalar uchun berilgan funksiya va interval uchun Rol teoremasining xulosasini qanoatlantiradigan barcha c son(lar)ni aniqlang.

  1. \(\left[ {0,4} \right]\) da \(f\left( x \right) = {x^3} - 4{x^2} + 3\)
  2. \(\left[ { - 2,4} \right]\) da \(Q\left( z \right) = 15 + 2z - {z^2}\)
  3. \(\left[ { - 3,3} \right]\) da \(h\left( t \right) = 1 - {{\bf{e}}^{{t^{\,2}} - 9}}\)
  4. \(\left[ {5,9} \right]\) da \(g\left( w \right) = 1 + \cos \left[ {\pi \,w} \right]\)

5 – 8 masalalar uchun berilgan funksiya va interval uchun o‘rtacha qiymat teoremasining xulosasini qanoatlantiradigan barcha c son(lar)ni aniqlang.

  1. \(\left[ { - 3,4} \right]\) da \(f\left( x \right) = {x^3} - {x^2} + x + 8\)
  2. \(g\left( t \right) = 2{t^3} + {t^2} + 7t - 1\), \(\left[ {1,6} \right]\)
  3. \(P\left( t \right) = {{\bf{e}}^{2t}} - 6t - 3\), \(\left[ { - 1,0} \right]\)
  4. \(h\left( x \right) = 9x - 8\sin \left( {{\displaystyle \frac{x}{2}}} \right)\), \(\left[ { - 3, - 1} \right]\)
  5. Faraz qilaylik, \(f\left( x \right)\) uzluksiz va \(\left[ { - 2,5} \right]\), bu \(f\left( 5 \right) = 14\) va \(f'\left( x \right) \le 10\) oralig‘ida differensiallanadi. \(f\left( { - 2} \right)\) uchun mumkin bo'lgan eng kichik qiymat nima?
  6. Faraz qilaylik, biz \(f\left( x \right)\) uzluksiz va \(\left[ { - 6, - 1} \right]\), bu \(f\left( { - 6} \right) = - 23\) va \(f'\left( x \right) \ge - 4\) oralig‘ida differensiallanuvchi ekanligini bilamiz. \(f\left( { - 1} \right)\) uchun mumkin bo'lgan eng kichik qiymat nima?
  7. Faraz qilaylik, \(f\left( x \right)\) uzluksiz va \(\left[ { - 3,4} \right]\), bu \(f\left( { - 3} \right) = 7\) va \(f'\left( x \right) \le - 17\) oralig‘ida differensiallanadi. \(f\left( 4 \right)\) uchun mumkin bo'lgan eng katta qiymat nima?
  8. Faraz qilaylik, \(f\left( x \right)\) uzluksiz va \(\left[ {1,9} \right]\), bu \(f\left( 9 \right) = 0\) va \(f'\left( x \right) \ge 8\) oralig‘ida differensial bo‘ladi. \(f\left( 1 \right)\) uchun mumkin bo'lgan eng katta qiymat nima?
  9. \(f\left( x \right) = {x^7} + 2{x^5} + 3{x^3} + 14x + 1\)ning aynan bitta haqiqiy ildizga ega ekanligini ko'rsating.
  10. \(f\left( x \right) = 6{x^3} - 2{x^2} + 4x - 3\) aynan bitta haqiqiy ildizga ega ekanligini ko'rsating.
  11. \(f\left( x \right) = 20x - {{\bf{e}}^{ - 4x}}\) aynan bitta haqiqiy ildizga ega ekanligini ko'rsating.