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### Section 4-7 : The Mean Value Theorem

For problems 1 & 2 determine all the number(s) c which satisfy the conclusion of Rolle’s Theorem for the given function and interval.

1. $$f\left( x \right) = {x^2} - 2x - 8$$ on $$\left[ { - 1,3} \right]$$ Solution
2. $$g\left( t \right) = 2t - {t^2} - {t^3}$$ on $$\left[ { - 2,1} \right]$$ Solution

For problems 3 & 4 determine all the number(s) c which satisfy the conclusion of the Mean Value Theorem for the given function and interval.

1. $$h\left( z \right) = 4{z^3} - 8{z^2} + 7z - 2$$ on $$\left[ {2,5} \right]$$ Solution
2. $$A\left( t \right) = 8t + {{\bf{e}}^{ - 3\,t}}$$ on $$\left[ { - 2,3} \right]$$ Solution
3. Suppose we know that $$f\left( x \right)$$ is continuous and differentiable on the interval $$\left[ { - 7,0} \right]$$, that $$f\left( { - 7} \right) = - 3$$ and that $$f'\left( x \right) \le 2$$. What is the largest possible value for $$f\left( 0 \right)$$? Solution
4. Show that $$f\left( x \right) = {x^3} - 7{x^2} + 25x + 8$$ has exactly one real root. Solution