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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.