Calculus I - The Mean Value Theorem (Practice Problems)

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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.

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

$h\left( z \right) = 4{z^3} - 8{z^2} + 7z - 2$ on $\left[ {2,5} \right]$ Solution
$A\left( t \right) = 8t + {{\bf{e}}^{ - 3\,t}}$ on $\left[ { - 2,3} \right]$ Solution
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
Show that $f\left( x \right) = {x^3} - 7{x^2} + 25x + 8$ has exactly one real root. Solution