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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