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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
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.
- \(f\left( x \right) = {x^3} - 4{x^2} + 3\) on \(\left[ {0,4} \right]\)
- \(Q\left( z \right) = 15 + 2z - {z^2}\) on \(\left[ { - 2,4} \right]\)
- \(h\left( t \right) = 1 - {{\bf{e}}^{{t^{\,2}} - 9}}\) on \(\left[ { - 3,3} \right]\)
- \(g\left( w \right) = 1 + \cos \left[ {\pi \,w} \right]\) 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.
- \(f\left( x \right) = {x^3} - {x^2} + x + 8\) on \(\left[ { - 3,4} \right]\)
- \(g\left( t \right) = 2{t^3} + {t^2} + 7t - 1\) on \(\left[ {1,6} \right]\)
- \(P\left( t \right) = {{\bf{e}}^{2t}} - 6t - 3\) on \(\left[ { - 1,0} \right]\)
- \(h\left( x \right) = 9x - 8\sin \left( {{\displaystyle \frac{x}{2}}} \right)\) on \(\left[ { - 3, - 1} \right]\)
- 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 \(f\left( { - 2} \right)\)?
- 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 \(f\left( { - 1} \right)\)?
- 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 \(f\left( 4 \right)\)?
- 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 \(f\left( 1 \right)\)?
- Show that \(f\left( x \right) = {x^7} + 2{x^5} + 3{x^3} + 14x + 1\) has exactly one real root.
- Show that \(f\left( x \right) = 6{x^3} - 2{x^2} + 4x - 3\) has exactly one real root.
- Show that \(f\left( x \right) = 20x - {{\bf{e}}^{ - 4x}}\) has exactly one real root.