Mean value theorem

Lessons

1. Consider the function $f\left( x \right) = 2{x^3} - 5x - 1$ on the interval [0, 2].
   
   1. Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval.
   2. Find all numbers between 0 and 2 that satisfy the conclusion of the Mean Value Theorem.
2. If $f\left( 2 \right) = 5$ and $f'\left( x \right) \ge 3$ for $2 \le x \le 10$, what is the smallest possible value for $f\left( {10} \right)?$