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Section 4-7 : The Mean Value Theorem

For problems 1 – 4 determine all the number(s) c which satisfy the conclusion of Rolle’s Theorem for the given function and interval.

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

For problems 5 – 8 determine all the number(s) c which satisfy the conclusion of the Mean Value Theorem for the given function and interval.

  1. \(f\left( x \right) = {x^3} - {x^2} + x + 8\) on \(\left[ { - 3,4} \right]\)
  2. \(g\left( t \right) = 2{t^3} + {t^2} + 7t - 1\) on \(\left[ {1,6} \right]\)
  3. \(P\left( t \right) = {{\bf{e}}^{2t}} - 6t - 3\) on \(\left[ { - 1,0} \right]\)
  4. \(h\left( x \right) = 9x - 8\sin \left( {{\displaystyle \frac{x}{2}}} \right)\) on \(\left[ { - 3, - 1} \right]\)
  5. Suppose we know that \(f\left( x \right)\) is continuous and differentiable on the interval \(\left[ { - 2,5} \right]\), that \(f\left( 5 \right) = 14\) and that \(f'\left( x \right) \le 10\). What is the smallest possible value for \(f\left( { - 2} \right)\)?
  6. Suppose we know that \(f\left( x \right)\) is continuous and differentiable on the interval \(\left[ { - 6, - 1} \right]\), that \(f\left( { - 6} \right) = - 23\) and that \(f'\left( x \right) \ge - 4\). What is the smallest possible value for \(f\left( { - 1} \right)\)?
  7. Suppose we know that \(f\left( x \right)\) is continuous and differentiable on the interval \(\left[ { - 3,4} \right]\), that \(f\left( { - 3} \right) = 7\) and that \(f'\left( x \right) \le - 17\). What is the largest possible value for \(f\left( 4 \right)\)?
  8. Suppose we know that \(f\left( x \right)\) is continuous and differentiable on the interval \(\left[ {1,9} \right]\), that \(f\left( 9 \right) = 0\) and that \(f'\left( x \right) \ge 8\). What is the largest possible value for \(f\left( 1 \right)\)?
  9. Show that \(f\left( x \right) = {x^7} + 2{x^5} + 3{x^3} + 14x + 1\) has exactly one real root.
  10. Show that \(f\left( x \right) = 6{x^3} - 2{x^2} + 4x - 3\) has exactly one real root.
  11. Show that \(f\left( x \right) = 20x - {{\bf{e}}^{ - 4x}}\) has exactly one real root.